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The finite element method (FEM) is a popular method for numerically solving
differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s arising in engineering and
mathematical modeling A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in the natural sciences (such as physics, b ...
. Typical problem areas of interest include the traditional fields of structural analysis,
heat transfer Heat transfer is a discipline of thermal engineering that concerns the generation, use, conversion, and exchange of thermal energy (heat) between physical systems. Heat transfer is classified into various mechanisms, such as thermal conduction, ...
, fluid flow, mass transport, and
electromagnetic potential An electromagnetic four-potential is a relativistic vector function from which the electromagnetic field can be derived. It combines both an electric scalar potential and a magnetic vector potential into a single four-vector.Gravitation, J.A. ...
. The FEM is a general numerical method for solving
partial differential equations In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to ...
in two or three space variables (i.e., some boundary value problems). To solve a problem, the FEM subdivides a large system into smaller, simpler parts that are called finite elements. This is achieved by a particular space discretization in the space dimensions, which is implemented by the construction of a
mesh A mesh is a barrier made of connected strands of metal, fiber, or other flexible or ductile materials. A mesh is similar to a web or a net in that it has many attached or woven strands. Types * A plastic mesh may be extruded, oriented, exp ...
of the object: the numerical domain for the solution, which has a finite number of points. The finite element method formulation of a boundary value problem finally results in a system of
algebraic equation In mathematics, an algebraic equation or polynomial equation is an equation of the form :P = 0 where ''P'' is a polynomial with coefficients in some field, often the field of the rational numbers. For many authors, the term ''algebraic equation'' ...
s. The method approximates the unknown function over the domain. The simple equations that model these finite elements are then assembled into a larger system of equations that models the entire problem. The FEM then approximates a solution by minimizing an associated error function via the calculus of variations. Studying or
analyzing Analysis (plural, : analyses) is the process of breaking a complexity, complex topic or Substance theory, substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics a ...
a phenomenon with FEM is often referred to as finite element analysis (FEA).


Basic concepts

The subdivision of a whole domain into simpler parts has several advantages: * Accurate representation of complex geometry * Inclusion of dissimilar material properties * Easy representation of the total solution * Capture of local effects. Typical work out of the method involves: # dividing the domain of the problem into a collection of subdomains, with each subdomain represented by a set of element equations to the original problem # systematically recombining all sets of element equations into a global system of equations for the final calculation. The global system of equations has known solution techniques, and can be calculated from the
initial value In multivariable calculus, an initial value problem (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. Modeling a system in physics or ot ...
s of the original problem to obtain a numerical answer. In the first step above, the element equations are simple equations that locally approximate the original complex equations to be studied, where the original equations are often partial differential equations (PDE). To explain the approximation in this process, the finite element method is commonly introduced as a special case of
Galerkin method In mathematics, in the area of numerical analysis, Galerkin methods, named after the Russian mathematician Boris Galerkin, convert a continuous operator problem, such as a differential equation, commonly in a weak formulation, to a discrete prob ...
. The process, in mathematical language, is to construct an integral of the
inner product In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
of the residual and the
weight function A weight function is a mathematical device used when performing a sum, integral, or average to give some elements more "weight" or influence on the result than other elements in the same set. The result of this application of a weight function is ...
s and set the integral to zero. In simple terms, it is a procedure that minimizes the error of approximation by fitting trial functions into the PDE. The residual is the error caused by the trial functions, and the weight functions are
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example ...
approximation functions that project the residual. The process eliminates all the spatial derivatives from the PDE, thus approximating the PDE locally with * a set of
algebraic equations In mathematics, an algebraic equation or polynomial equation is an equation of the form :P = 0 where ''P'' is a polynomial with coefficients in some field, often the field of the rational numbers. For many authors, the term ''algebraic equation'' ...
for
steady state In systems theory, a system or a process is in a steady state if the variables (called state variables) which define the behavior of the system or the process are unchanging in time. In continuous time, this means that for those properties ''p' ...
problems, * a set of
ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...
s for transient problems. These equation sets are the element equations. They are
linear Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
if the underlying PDE is linear, and vice versa. Algebraic equation sets that arise in the steady-state problems are solved using
numerical linear algebra Numerical linear algebra, sometimes called applied linear algebra, is the study of how matrix operations can be used to create computer algorithms which efficiently and accurately provide approximate answers to questions in continuous mathematic ...
methods, while
ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...
sets that arise in the transient problems are solved by numerical integration using standard techniques such as
Euler's method In mathematics and computational science, the Euler method (also called forward Euler method) is a first-order numerical procedure for solving ordinary differential equations (ODEs) with a given initial value. It is the most basic explicit met ...
or the Runge-Kutta method. In step (2) above, a global system of equations is generated from the element equations through a transformation of coordinates from the subdomains' local nodes to the domain's global nodes. This spatial transformation includes appropriate orientation adjustments as applied in relation to the reference coordinate system. The process is often carried out by FEM software using
coordinate In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sign ...
data generated from the subdomains. The practical application of FEM is known as ''finite element analysis'' (FEA). FEA as applied in
engineering Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...
is a computational tool for performing
engineering analysis Engineering analysis involves the application of scientific/mathematical analytic principles and processes to reveal the properties and state of a system, device or mechanism under study. Engineering analysis is decompositional: it proceeds by se ...
. It includes the use of
mesh generation Mesh generation is the practice of creating a mesh, a subdivision of a continuous geometric space into discrete geometric and topological cells. Often these cells form a simplicial complex. Usually the cells partition the geometric input domain. ...
techniques for dividing a complex problem into small elements, as well as the use of software coded with a FEM algorithm. In applying FEA, the complex problem is usually a physical system with the underlying
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
such as the Euler–Bernoulli beam equation, the heat equation, or the Navier-Stokes equations expressed in either PDE or
integral equation In mathematics, integral equations are equations in which an unknown function appears under an integral sign. In mathematical notation, integral equations may thus be expressed as being of the form: f(x_1,x_2,x_3,...,x_n ; u(x_1,x_2,x_3,...,x_n) ...
s, while the divided small elements of the complex problem represent different areas in the physical system. FEA may be used for analyzing problems over complicated domains (like cars and oil pipelines), when the domain changes (as during a solid-state reaction with a moving boundary), when the desired precision varies over the entire domain, or when the solution lacks smoothness. FEA simulations provide a valuable resource as they remove multiple instances of creation and testing of hard prototypes for various high fidelity situations. For instance, in a frontal crash simulation it is possible to increase prediction accuracy in "important" areas like the front of the car and reduce it in its rear (thus reducing the cost of the simulation). Another example would be in
numerical weather prediction Numerical weather prediction (NWP) uses mathematical models of the atmosphere and oceans to predict the weather based on current weather conditions. Though first attempted in the 1920s, it was not until the advent of computer simulation in th ...
, where it is more important to have accurate predictions over developing highly nonlinear phenomena (such as
tropical cyclone A tropical cyclone is a rapidly rotating storm system characterized by a low-pressure center, a closed low-level atmospheric circulation, strong winds, and a spiral arrangement of thunderstorms that produce heavy rain and squalls. Depen ...
s in the atmosphere, or
eddies In fluid dynamics, an eddy is the swirling of a fluid and the reverse current created when the fluid is in a turbulent flow regime. The moving fluid creates a space devoid of downstream-flowing fluid on the downstream side of the object. Fluid b ...
in the ocean) rather than relatively calm areas. A clear, detailed and practical presentation of this approach can be found in ''The Finite Element Method for Engineers''.


History

While it is difficult to quote a date of the invention of the finite element method, the method originated from the need to solve complex elasticity and structural analysis problems in
civil Civil may refer to: *Civic virtue, or civility *Civil action, or lawsuit * Civil affairs *Civil and political rights *Civil disobedience *Civil engineering *Civil (journalism), a platform for independent journalism *Civilian, someone not a membe ...
and
aeronautical engineering Aerospace engineering is the primary field of engineering concerned with the development of aircraft and spacecraft. It has two major and overlapping branches: aeronautical engineering and astronautical engineering. Avionics engineering is sim ...
. Its development can be traced back to the work by A. Hrennikoff and R. Courant in the early 1940s. Another pioneer was Ioannis Argyris. In the USSR, the introduction of the practical application of the method is usually connected with name of
Leonard Oganesyan Leonard or ''Leo'' is a common English masculine given name and a surname. The given name and surname originate from the Old High German ''Leonhard'' containing the prefix ''levon'' ("lion") from the Greek Λέων ("lion") through the Latin '' L ...
. It was also independently rediscovered in China by Feng Kang in the later 1950s and early 1960s, based on the computations of dam constructions, where it was called the ''finite difference method based on variation principle''. Although the approaches used by these pioneers are different, they share one essential characteristic:
mesh A mesh is a barrier made of connected strands of metal, fiber, or other flexible or ductile materials. A mesh is similar to a web or a net in that it has many attached or woven strands. Types * A plastic mesh may be extruded, oriented, exp ...
discretization of a continuous domain into a set of discrete sub-domains, usually called elements. Hrennikoff's work discretizes the domain by using a
lattice Lattice may refer to: Arts and design * Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material * Lattice (music), an organized grid model of pitch ratios * Lattice (pastry), an orna ...
analogy, while Courant's approach divides the domain into finite triangular subregions to solve second order elliptic partial differential equations that arise from the problem of
torsion Torsion may refer to: Science * Torsion (mechanics), the twisting of an object due to an applied torque * Torsion of spacetime, the field used in Einstein–Cartan theory and ** Alternatives to general relativity * Torsion angle, in chemistry Bi ...
of a
cylinder A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base. A cylinder may also be defined as an infin ...
. Courant's contribution was evolutionary, drawing on a large body of earlier results for PDEs developed by Rayleigh, Ritz, and Galerkin. The finite element method obtained its real impetus in the 1960s and 1970s by the developments of J. H. Argyris with co-workers at the University of Stuttgart, R. W. Clough with co-workers at
UC Berkeley The University of California, Berkeley (UC Berkeley, Berkeley, Cal, or California) is a public land-grant research university in Berkeley, California. Established in 1868 as the University of California, it is the state's first land-grant uni ...
, O. C. Zienkiewicz with co-workers Ernest Hinton, Bruce Irons and others at
Swansea University Swansea University ( cy, Prifysgol Abertawe) is a public university, public research university located in Swansea, Wales, United Kingdom. It was chartered as University College of Swansea in 1920, as the fourth college of the University of Wales. ...
, Philippe G. Ciarlet at the University of Paris 6 and Richard Gallagher with co-workers at
Cornell University Cornell University is a private statutory land-grant research university based in Ithaca, New York. It is a member of the Ivy League. Founded in 1865 by Ezra Cornell and Andrew Dickson White, Cornell was founded with the intention to tea ...
. Further impetus was provided in these years by available open source finite element programs. NASA sponsored the original version of
NASTRAN NASTRAN is a finite element analysis (FEA) program that was originally developed for NASA in the late 1960s under United States government funding for the aerospace industry. The MacNeal-Schwendler Corporation (MSC) was one of the principal and o ...
, and UC Berkeley made the finite element program SAP IV widely available. In Norway the ship classification society Det Norske Veritas (now
DNV GL DNV (formerly DNV GL) is an international accredited registrar and classification society headquartered in Høvik, Norway. The company currently has about 12,000 employees and 350 offices operating in more than 100 countries, and provides serv ...
) developed
Sesam Sesam, SESAM or SeSaM may refer to: * SESAM (database), a relational database developed by Fujitsu Siemens * SESAM (FEM), a structural analysis software * Sesam (search engine), a Scandinavian internet search engine * SeSaM-Biotech GmbH, a biotech ...
in 1969 for use in analysis of ships. A rigorous mathematical basis to the finite element method was provided in 1973 with the publication by Strang and Fix. The method has since been generalized for the numerical modeling of physical systems in a wide variety of
engineering Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...
disciplines, e.g.,
electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of ...
,
heat transfer Heat transfer is a discipline of thermal engineering that concerns the generation, use, conversion, and exchange of thermal energy (heat) between physical systems. Heat transfer is classified into various mechanisms, such as thermal conduction, ...
, and fluid dynamics.


Technical discussion


The structure of finite element methods

A finite element method is characterized by a variational formulation, a discretization strategy, one or more solution algorithms, and post-processing procedures. Examples of the variational formulation are the
Galerkin method In mathematics, in the area of numerical analysis, Galerkin methods, named after the Russian mathematician Boris Galerkin, convert a continuous operator problem, such as a differential equation, commonly in a weak formulation, to a discrete prob ...
, the discontinuous Galerkin method, mixed methods, etc. A discretization strategy is understood to mean a clearly defined set of procedures that cover (a) the creation of finite element meshes, (b) the definition of basis function on reference elements (also called shape functions) and (c) the mapping of reference elements onto the elements of the mesh. Examples of discretization strategies are the h-version, p-version, hp-version, x-FEM, isogeometric analysis, etc. Each discretization strategy has certain advantages and disadvantages. A reasonable criterion in selecting a discretization strategy is to realize nearly optimal performance for the broadest set of mathematical models in a particular model class. Various numerical solution algorithms can be classified into two broad categories; direct and iterative solvers. These algorithms are designed to exploit the sparsity of matrices that depend on the choices of variational formulation and discretization strategy. Postprocessing procedures are designed for the extraction of the data of interest from a finite element solution. In order to meet the requirements of solution verification, postprocessors need to provide for ''a posteriori'' error estimation in terms of the quantities of interest. When the errors of approximation are larger than what is considered acceptable then the discretization has to be changed either by an automated adaptive process or by the action of the analyst. There are some very efficient postprocessors that provide for the realization of superconvergence.


Illustrative problems P1 and P2

The following two problems demonstrate the finite element method. P1 is a one-dimensional problem :\mbox:\begin u''(x)=f(x) \mbox (0,1), \\ u(0)=u(1)=0, \end where f is given, u is an unknown function of x, and u'' is the second derivative of u with respect to x. P2 is a two-dimensional problem (
Dirichlet problem In mathematics, a Dirichlet problem is the problem of finding a function which solves a specified partial differential equation (PDE) in the interior of a given region that takes prescribed values on the boundary of the region. The Dirichlet prob ...
) :\mbox:\begin u_(x,y)+u_(x,y)=f(x,y) & \mbox \Omega, \\ u=0 & \mbox \partial \Omega, \end where \Omega is a connected open region in the (x,y) plane whose boundary \partial \Omega is nice (e.g., a
smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
or a
polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two to ...
), and u_ and u_ denote the second derivatives with respect to x and y, respectively. The problem P1 can be solved directly by computing
antiderivative In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolicall ...
s. However, this method of solving the boundary value problem (BVP) works only when there is one spatial dimension and does not generalize to higher-dimensional problems or problems like u+u''=f. For this reason, we will develop the finite element method for P1 and outline its generalization to P2. Our explanation will proceed in two steps, which mirror two essential steps one must take to solve a boundary value problem (BVP) using the FEM. * In the first step, one rephrases the original BVP in its weak form. Little to no computation is usually required for this step. The transformation is done by hand on paper. * The second step is the discretization, where the weak form is discretized in a finite-dimensional space. After this second step, we have concrete formulae for a large but finite-dimensional linear problem whose solution will approximately solve the original BVP. This finite-dimensional problem is then implemented on a computer.


Weak formulation

The first step is to convert P1 and P2 into their equivalent
weak formulation Weak formulations are important tools for the analysis of mathematical equations that permit the transfer of concepts of linear algebra to solve problems in other fields such as partial differential equations. In a weak formulation, equations or con ...
s.


The weak form of P1

If u solves P1, then for any smooth function v that satisfies the displacement boundary conditions, i.e. v=0 at x=0 and x=1, we have Conversely, if u with u(0) = u(1) = 0 satisfies (1) for every smooth function v(x) then one may show that this u will solve P1. The proof is easier for twice continuously differentiable u ( mean value theorem), but may be proved in a distributional sense as well. We define a new operator or map \phi(u,v) by using
integration by parts In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivative. ...
on the right-hand-side of (1): where we have used the assumption that v(0) = v(1) = 0.


The weak form of P2

If we integrate by parts using a form of Green's identities, we see that if u solves P2, then we may define \phi(u,v) for any v by :\int_\Omega fv\,ds = -\int_\Omega \nabla u \cdot \nabla v \, ds \equiv -\phi(u,v), where \nabla denotes the
gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
and \cdot denotes the
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alge ...
in the two-dimensional plane. Once more \,\!\phi can be turned into an inner product on a suitable space H_0^1(\Omega) of once differentiable functions of \Omega that are zero on \partial \Omega. We have also assumed that v \in H_0^1(\Omega) (see
Sobolev space In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense ...
s). Existence and uniqueness of the solution can also be shown.


A proof outline of existence and uniqueness of the solution

We can loosely think of H_0^1(0,1) to be the
absolutely continuous In calculus, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations of the relationship between the two central ope ...
functions of (0,1) that are 0 at x=0 and x=1 (see
Sobolev spaces In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense ...
). Such functions are (weakly) once differentiable and it turns out that the symmetric
bilinear map In mathematics, a bilinear map is a function combining elements of two vector spaces to yield an element of a third vector space, and is linear in each of its arguments. Matrix multiplication is an example. Definition Vector spaces Let V, W ...
\!\,\phi then defines an
inner product In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
which turns H_0^1(0,1) into a Hilbert space (a detailed proof is nontrivial). On the other hand, the left-hand-side \int_0^1 f(x)v(x)dx is also an inner product, this time on the Lp space L^2(0,1). An application of the
Riesz representation theorem :''This article describes a theorem concerning the dual of a Hilbert space. For the theorems relating linear functionals to Measure (mathematics), measures, see Riesz–Markov–Kakutani representation theorem.'' The Riesz representation theorem, ...
for Hilbert spaces shows that there is a unique u solving (2) and therefore P1. This solution is a-priori only a member of H_0^1(0,1), but using
elliptic In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse in ...
regularity, will be smooth if f is.


Discretization

P1 and P2 are ready to be discretized which leads to a common sub-problem (3). The basic idea is to replace the infinite-dimensional linear problem: :Find u \in H_0^1 such that :\forall v \in H_0^1, \; -\phi(u,v)=\int fv with a finite-dimensional version: where V is a finite-dimensional subspace of H_0^1. There are many possible choices for V (one possibility leads to the
spectral method Spectral methods are a class of techniques used in applied mathematics and scientific computing to numerically solve certain differential equations. The idea is to write the solution of the differential equation as a sum of certain "basis function ...
). However, for the finite element method we take V to be a space of piecewise polynomial functions.


For problem P1

We take the interval (0,1), choose n values of x with 0=x_0 < x_1 < \cdots < x_n < x_=1 and we define V by: :V=\ where we define x_0=0 and x_=1. Observe that functions in V are not differentiable according to the elementary definition of calculus. Indeed, if v \in V then the derivative is typically not defined at any x=x_k, k=1,\ldots,n. However, the derivative exists at every other value of x and one can use this derivative for the purpose of
integration by parts In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivative. ...
.


For problem P2

We need V to be a set of functions of \Omega. In the figure on the right, we have illustrated a triangulation of a 15 sided
polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two to ...
al region \Omega in the plane (below), and a piecewise linear function (above, in color) of this polygon which is linear on each triangle of the triangulation; the space V would consist of functions that are linear on each triangle of the chosen triangulation. One hopes that as the underlying triangular mesh becomes finer and finer, the solution of the discrete problem (3) will in some sense converge to the solution of the original boundary value problem P2. To measure this mesh fineness, the triangulation is indexed by a real-valued parameter h > 0 which one takes to be very small. This parameter will be related to the size of the largest or average triangle in the triangulation. As we refine the triangulation, the space of piecewise linear functions V must also change with h. For this reason, one often reads V_h instead of V in the literature. Since we do not perform such an analysis, we will not use this notation.


Choosing a basis

To complete the discretization, we must select a
basis Basis may refer to: Finance and accounting * Adjusted basis, the net cost of an asset after adjusting for various tax-related items *Basis point, 0.01%, often used in the context of interest rates * Basis trading, a trading strategy consisting ...
of V. In the one-dimensional case, for each control point x_k we will choose the piecewise linear function v_k in V whose value is 1 at x_k and zero at every x_j,\;j \neq k, i.e., :v_(x)=\begin & \mbox x \in _,x_k \\ & \mbox x \in _k,x_ \\ 0 & \mbox, \end for k=1,\dots,n; this basis is a shifted and scaled tent function. For the two-dimensional case, we choose again one basis function v_k per vertex x_k of the triangulation of the planar region \Omega. The function v_k is the unique function of V whose value is 1 at x_k and zero at every x_j,\;j \neq k. Depending on the author, the word "element" in the "finite element method" refers either to the triangles in the domain, the piecewise linear basis function, or both. So for instance, an author interested in curved domains might replace the triangles with curved primitives, and so might describe the elements as being curvilinear. On the other hand, some authors replace "piecewise linear" by "piecewise quadratic" or even "piecewise polynomial". The author might then say "higher order element" instead of "higher degree polynomial". The finite element method is not restricted to triangles (or tetrahedra in 3-d, or higher-order simplexes in multidimensional spaces), but can be defined on quadrilateral subdomains (hexahedra, prisms, or pyramids in 3-d, and so on). Higher-order shapes (curvilinear elements) can be defined with polynomial and even non-polynomial shapes (e.g. ellipse or circle). Examples of methods that use higher degree piecewise polynomial basis functions are the
hp-FEM hp-FEM is a general version of the finite element method (FEM), a numerical method for solving partial differential equations based on piecewise-polynomial approximations that employs elements of variable size ''(h)'' and polynomial degree ' ...
and spectral FEM. More advanced implementations (adaptive finite element methods) utilize a method to assess the quality of the results (based on error estimation theory) and modify the mesh during the solution aiming to achieve an approximate solution within some bounds from the exact solution of the continuum problem. Mesh adaptivity may utilize various techniques, the most popular are: * moving nodes (r-adaptivity) * refining (and unrefined) elements (h-adaptivity) * changing order of base functions (p-adaptivity) * combinations of the above ( hp-adaptivity).


Small support of the basis

The primary advantage of this choice of basis is that the inner products :\langle v_j,v_k \rangle=\int_0^1 v_j v_k\,dx and :\phi(v_j,v_k)=\int_0^1 v_j' v_k'\,dx will be zero for almost all j,k. (The matrix containing \langle v_j,v_k \rangle in the (j,k) location is known as the
Gramian matrix In linear algebra, the Gram matrix (or Gramian matrix, Gramian) of a set of vectors v_1,\dots, v_n in an inner product space is the Hermitian matrix of inner products, whose entries are given by the inner product G_ = \left\langle v_i, v_j \right\r ...
.) In the one dimensional case, the support of v_k is the interval _,x_/math>. Hence, the integrands of \langle v_j,v_k \rangle and ''\phi(v_j,v_k)'' are identically zero whenever , j-k, >1. Similarly, in the planar case, if x_j and x_k do not share an edge of the triangulation, then the integrals :\int_ v_j v_k\,ds and :\int_ \nabla v_j \cdot \nabla v_k\,ds are both zero.


Matrix form of the problem

If we write u(x)=\sum_^n u_k v_k(x) and f(x)=\sum_^n f_k v_k(x) then problem (3), taking v(x)=v_j(x) for j=1,\dots,n, becomes If we denote by \mathbf and \mathbf the column vectors (u_1,\dots,u_n)^t and (f_1,\dots,f_n)^t, and if we let :L=(L_) and :M=(M_) be matrices whose entries are :L_=\phi (v_i,v_j) and :M_=\int v_i v_j dx then we may rephrase (4) as It is not necessary to assume f(x)=\sum_^n f_k v_k(x). For a general function f(x), problem (3) with v(x)=v_j(x) for j=1,\dots,n becomes actually simpler, since no matrix M is used, where \mathbf=(b_1,\dots,b_n)^t and b_j=\int f v_j dx for j=1,\dots,n. As we have discussed before, most of the entries of L and M are zero because the basis functions v_k have small support. So we now have to solve a linear system in the unknown \mathbf where most of the entries of the matrix L, which we need to invert, are zero. Such matrices are known as
sparse matrices In numerical analysis and scientific computing, a sparse matrix or sparse array is a matrix in which most of the elements are zero. There is no strict definition regarding the proportion of zero-value elements for a matrix to qualify as sparse b ...
, and there are efficient solvers for such problems (much more efficient than actually inverting the matrix.) In addition, L is symmetric and positive definite, so a technique such as the conjugate gradient method is favored. For problems that are not too large, sparse
LU decomposition In numerical analysis and linear algebra, lower–upper (LU) decomposition or factorization factors a matrix as the product of a lower triangular matrix and an upper triangular matrix (see matrix decomposition). The product sometimes includes a p ...
s and
Cholesky decomposition In linear algebra, the Cholesky decomposition or Cholesky factorization (pronounced ) is a decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose, which is useful for effici ...
s still work well. For instance,
MATLAB MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementa ...
's backslash operator (which uses sparse LU, sparse Cholesky, and other factorization methods) can be sufficient for meshes with a hundred thousand vertices. The matrix L is usually referred to as the
stiffness matrix In the finite element method for the numerical solution of elliptic partial differential equations, the stiffness matrix is a matrix that represents the system of linear equations that must be solved in order to ascertain an approximate solution ...
, while the matrix M is dubbed the
mass matrix In analytical mechanics, the mass matrix is a symmetric matrix that expresses the connection between the time derivative \mathbf\dot q of the generalized coordinate vector of a system and the kinetic energy of that system, by the equation :T ...
.


General form of the finite element method

In general, the finite element method is characterized by the following process. *One chooses a grid for \Omega. In the preceding treatment, the grid consisted of triangles, but one can also use squares or curvilinear polygons. *Then, one chooses basis functions. In our discussion, we used piecewise linear basis functions, but it is also common to use piecewise polynomial basis functions. Separate consideration is the smoothness of the basis functions. For second-order
elliptic boundary value problem In mathematics, an elliptic boundary value problem is a special kind of boundary value problem which can be thought of as the stable state of an evolution problem. For example, the Dirichlet problem for the Laplacian gives the eventual distri ...
s, piecewise polynomial basis function that is merely continuous suffice (i.e., the derivatives are discontinuous.) For higher-order partial differential equations, one must use smoother basis functions. For instance, for a fourth-order problem such as u_+u_=f, one may use piecewise quadratic basis functions that are C^1. Another consideration is the relation of the finite-dimensional space V to its infinite-dimensional counterpart, in the examples above H_0^1. A conforming element method is one in which space V is a subspace of the element space for the continuous problem. The example above is such a method. If this condition is not satisfied, we obtain a nonconforming element method, an example of which is the space of piecewise linear functions over the mesh which are continuous at each edge midpoint. Since these functions are in general discontinuous along the edges, this finite-dimensional space is not a subspace of the original H_0^1. Typically, one has an algorithm for taking a given mesh and subdividing it. If the main method for increasing precision is to subdivide the mesh, one has an ''h''-method (''h'' is customarily the diameter of the largest element in the mesh.) In this manner, if one shows that the error with a grid h is bounded above by Ch^p, for some C<\infty and p>0, then one has an order ''p'' method. Under certain hypotheses (for instance, if the domain is convex), a piecewise polynomial of order d method will have an error of order p=d+1. If instead of making ''h'' smaller, one increases the degree of the polynomials used in the basis function, one has a ''p''-method. If one combines these two refinement types, one obtains an ''hp''-method (
hp-FEM hp-FEM is a general version of the finite element method (FEM), a numerical method for solving partial differential equations based on piecewise-polynomial approximations that employs elements of variable size ''(h)'' and polynomial degree ' ...
). In the hp-FEM, the polynomial degrees can vary from element to element. High order methods with large uniform ''p'' are called spectral finite element methods ( SFEM). These are not to be confused with
spectral method Spectral methods are a class of techniques used in applied mathematics and scientific computing to numerically solve certain differential equations. The idea is to write the solution of the differential equation as a sum of certain "basis function ...
s. For vector partial differential equations, the basis functions may take values in \mathbb^n.


Various types of finite element methods


AEM

The Applied Element Method or AEM combines features of both FEM and
Discrete element method A discrete element method (DEM), also called a distinct element method, is any of a family of numerical methods for computing the motion and effect of a large number of small particles. Though DEM is very closely related to molecular dynamics, t ...
, or (DEM).


A-FEM

The Augmented-Finite Element Method is introduced by Yang and Lui whose goal was to model the weak and strong discontinuities without the need of extra DoFs as in PuM stated.


Generalized finite element method

The generalized finite element method (GFEM) uses local spaces consisting of functions, not necessarily polynomials, that reflect the available information on the unknown solution and thus ensure good local approximation. Then a
partition of unity In mathematics, a partition of unity of a topological space is a set of continuous functions from to the unit interval ,1such that for every point x\in X: * there is a neighbourhood of where all but a finite number of the functions of are 0 ...
is used to “bond” these spaces together to form the approximating subspace. The effectiveness of GFEM has been shown when applied to problems with domains having complicated boundaries, problems with micro-scales, and problems with boundary layers.


Mixed finite element method

The mixed finite element method is a type of finite element method in which extra independent variables are introduced as nodal variables during the discretization of a partial differential equation problem.


Variable – polynomial

The
hp-FEM hp-FEM is a general version of the finite element method (FEM), a numerical method for solving partial differential equations based on piecewise-polynomial approximations that employs elements of variable size ''(h)'' and polynomial degree ' ...
combines adaptively, elements with variable size ''h'' and polynomial degree ''p'' in order to achieve exceptionally fast, exponential convergence rates.


hpk-FEM

The hpk-FEM combines adaptively, elements with variable size ''h'', polynomial degree of the local approximations ''p'' and global differentiability of the local approximations ''(k-1)'' to achieve best convergence rates.


XFEM

The extended finite element method (XFEM) is a numerical technique based on the generalized finite element method (GFEM) and the partition of unity method (PUM). It extends the classical finite element method by enriching the solution space for solutions to differential equations with discontinuous functions. Extended finite element methods enrich the approximation space so that it can naturally reproduce the challenging feature associated with the problem of interest: the discontinuity, singularity, boundary layer, etc. It was shown that for some problems, such an embedding of the problem's feature into the approximation space can significantly improve convergence rates and accuracy. Moreover, treating problems with discontinuities with XFEMs suppresses the need to mesh and re-mesh the discontinuity surfaces, thus alleviating the computational costs and projection errors associated with conventional finite element methods, at the cost of restricting the discontinuities to mesh edges. Several research codes implement this technique to various degrees: 1. GetFEM++ 2. xfem++ 3. openxfem++ XFEM has also been implemented in codes like Altair Radios, ASTER, Morfeo, and Abaqus. It is increasingly being adopted by other commercial finite element software, with a few plugins and actual core implementations available (ANSYS, SAMCEF, OOFELIE, etc.).


Scaled boundary finite element method (SBFEM)

The introduction of the scaled boundary finite element method (SBFEM) came from Song and Wolf (1997). The SBFEM has been one of the most profitable contributions in the area of numerical analysis of fracture mechanics problems. It is a semi-analytical fundamental-solutionless method which combines the advantages of both the finite element formulations and procedures and the boundary element discretization. However, unlike the boundary element method, no fundamental differential solution is required.


S-FEM

The S-FEM, Smoothed Finite Element Methods, is a particular class of numerical simulation algorithms for the simulation of physical phenomena. It was developed by combining meshfree methods with the finite element method.


Spectral element method

Spectral element methods combine the geometric flexibility of finite elements and the acute accuracy of spectral methods. Spectral methods are the approximate solution of weak form partial equations that are based on high-order Lagrangian interpolants and used only with certain quadrature rules.


Meshfree methods


Discontinuous Galerkin methods


Finite element limit analysis


Stretched grid method


Loubignac iteration

Loubignac iteration In applied mathematics, Loubignac iteration is an iterative method in finite element method The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. ...
is an iterative method in finite element methods.


Crystal plasticity finite element method (CPFEM)

Crystal plasticity finite element method (CPFEM) is an advanced numerical tool developed by Franz Roters. Metals can be regarded as crystal aggregates and it behave anisotropy under deformation, for example, abnormal stress and strain localization. CPFEM based on slip (shear strain rate) can calculate dislocation, crystal orientation and other texture information to consider crystal anisotropy during the routine. Now it has been applied in the numerical study of material deformation, surface roughness, fractures and so on.


Virtual element method (VEM)

The virtual element method (VEM), introduced by Beirão da Veiga et al. (2013) as an extension of
mimetic Mimesis (; grc, μίμησις, ''mīmēsis'') is a term used in literary criticism and philosophy that carries a wide range of meanings, including '' imitatio'', imitation, nonsensuous similarity, receptivity, representation, mimicry, the ...
finite difference A finite difference is a mathematical expression of the form . If a finite difference is divided by , one gets a difference quotient. The approximation of derivatives by finite differences plays a central role in finite difference methods for t ...
(MFD) methods, is a generalisation of the standard finite element method for arbitrary element geometries. This allows admission of general polygons (or
polyhedra In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on ...
in 3D) that are highly irregular and non-convex in shape. The name ''virtual'' derives from the fact that knowledge of the local shape function basis is not required, and is in fact never explicitly calculated.


Link with the gradient discretization method

Some types of finite element methods (conforming, nonconforming, mixed finite element methods) are particular cases of the
gradient discretization method In numerical mathematics, the gradient discretisation method (GDM) is a framework which contains classical and recent numerical schemes for diffusion problems of various kinds: linear or non-linear, steady-state or time-dependent. The schemes m ...
(GDM). Hence the convergence properties of the GDM, which are established for a series of problems (linear and non-linear elliptic problems, linear, nonlinear, and degenerate parabolic problems), hold as well for these particular finite element methods.


Comparison to the finite difference method

The
finite difference method In numerical analysis, finite-difference methods (FDM) are a class of numerical techniques for solving differential equations by approximating derivatives with finite differences. Both the spatial domain and time interval (if applicable) are ...
(FDM) is an alternative way of approximating solutions of PDEs. The differences between FEM and FDM are: * The most attractive feature of the FEM is its ability to handle complicated geometries (and boundaries) with relative ease. While FDM in its basic form is restricted to handle rectangular shapes and simple alterations thereof, the handling of geometries in FEM is theoretically straightforward. * FDM is not usually used for irregular CAD geometries but more often rectangular or block shaped models. * FEM generally allows for more flexible mesh adaptivity than FDM. * The most attractive feature of finite differences is that it is very easy to implement. * There are several ways one could consider the FDM a special case of the FEM approach. E.g., first-order FEM is identical to FDM for
Poisson's equation Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with t ...
, if the problem is discretized by a regular rectangular mesh with each rectangle divided into two triangles. * There are reasons to consider the mathematical foundation of the finite element approximation more sound, for instance, because the quality of the approximation between grid points is poor in FDM. * The quality of a FEM approximation is often higher than in the corresponding FDM approach, but this is extremely problem-dependent and several examples to the contrary can be provided. Generally, FEM is the method of choice in all types of analysis in structural mechanics (i.e. solving for deformation and stresses in solid bodies or dynamics of structures) while
computational fluid dynamics Computational fluid dynamics (CFD) is a branch of fluid mechanics that uses numerical analysis and data structures to analyze and solve problems that involve fluid flows. Computers are used to perform the calculations required to simulate ...
(CFD) tend to use FDM or other methods like
finite volume method The finite volume method (FVM) is a method for representing and evaluating partial differential equations in the form of algebraic equations. In the finite volume method, volume integrals in a partial differential equation that contain a divergenc ...
(FVM). CFD problems usually require discretization of the problem into a large number of cells/gridpoints (millions and more), therefore the cost of the solution favors simpler, lower-order approximation within each cell. This is especially true for 'external flow' problems, like airflow around the car or airplane, or weather simulation.


Application

A variety of specializations under the umbrella of the mechanical engineering discipline (such as aeronautical, biomechanical, and automotive industries) commonly use integrated FEM in the design and development of their products. Several modern FEM packages include specific components such as thermal, electromagnetic, fluid, and structural working environments. In a structural simulation, FEM helps tremendously in producing stiffness and strength visualizations and also in minimizing weight, materials, and costs. FEM allows detailed visualization of where structures bend or twist, and indicates the distribution of stresses and displacements. FEM software provides a wide range of simulation options for controlling the complexity of both modeling and analysis of a system. Similarly, the desired level of accuracy required and associated computational time requirements can be managed simultaneously to address most engineering applications. FEM allows entire designs to be constructed, refined, and optimized before the design is manufactured. The mesh is an integral part of the model and it must be controlled carefully to give the best results. Generally the higher the number of elements in a mesh, the more accurate the solution of the discretized problem. However, there is a value at which the results converge and further mesh refinement does not increase accuracy. This powerful design tool has significantly improved both the standard of engineering designs and the methodology of the design process in many industrial applications.Hastings, J. K., Juds, M. A., Brauer, J. R., ''Accuracy and Economy of Finite Element Magnetic Analysis'', 33rd Annual National Relay Conference, April 1985. The introduction of FEM has substantially decreased the time to take products from concept to the production line. It is primarily through improved initial prototype designs using FEM that testing and development have been accelerated. In summary, benefits of FEM include increased accuracy, enhanced design and better insight into critical design parameters, virtual prototyping, fewer hardware prototypes, a faster and less expensive design cycle, increased productivity, and increased revenue. In the 1990s FEM was proposed for use in stochastic modelling for numerically solving probability models and later for reliability assessment.


See also

* Applied element method *
Boundary element method The boundary element method (BEM) is a numerical computational method of solving linear partial differential equations which have been formulated as integral equations (i.e. in ''boundary integral'' form), including fluid mechanics, acoustics, el ...
*
Céa's lemma Céa's lemma is a lemma in mathematics. Introduced by Jean Céa in his Ph.D. dissertation, it is an important tool for proving error estimates for the finite element method applied to elliptic partial differential equations. Lemma statement Let ...
*
Computer experiment A computer experiment or simulation experiment is an experiment used to study a computer simulation, also referred to as an in silico system. This area includes computational physics, computational chemistry, computational biology and other similar ...
* Direct stiffness method *
Discontinuity layout optimization Discontinuity layout optimization (DLO) is an engineering analysis procedure which can be used to directly establish the amount of load that can be carried by a solid or structure prior to collapse. Using DLO the layout of failure planes, or 'dis ...
*
Discrete element method A discrete element method (DEM), also called a distinct element method, is any of a family of numerical methods for computing the motion and effect of a large number of small particles. Though DEM is very closely related to molecular dynamics, t ...
*
Finite difference method In numerical analysis, finite-difference methods (FDM) are a class of numerical techniques for solving differential equations by approximating derivatives with finite differences. Both the spatial domain and time interval (if applicable) are ...
*
Finite element machine The Finite Element Machine (FEM) was a late 1970s-early 1980s NASA project to build and evaluate the performance of a parallel computer for structural analysis. The FEM was completed and successfully tested at the NASA Langley Research Center in H ...
*
Finite element method in structural mechanics The finite element method (FEM) is a powerful technique originally developed for numerical solution of complex problems in structural mechanics, and it remains the method of choice for complex systems. In the FEM, the structural system is modeled b ...
*
Finite volume method The finite volume method (FVM) is a method for representing and evaluating partial differential equations in the form of algebraic equations. In the finite volume method, volume integrals in a partial differential equation that contain a divergenc ...
*
Finite volume method for unsteady flow Unsteady flows are characterized as flows in which the properties of the fluid are time dependent. It gets reflected in the governing equations as the time derivative of the properties are absent. For Studying Finite-volume method for unsteady flow ...
* Infinite element method * Interval finite element * Isogeometric analysis *
Lattice Boltzmann methods The lattice Boltzmann methods (LBM), originated from the lattice gas automata (LGA) method (Hardy- Pomeau-Pazzis and Frisch- Hasslacher- Pomeau models), is a class of computational fluid dynamics (CFD) methods for fluid simulation. Instead of sol ...
*
List of finite element software packages This is a list of notable software packages that implement the finite element method for solving partial differential equations In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various ...
* Meshfree methods * Movable cellular automaton *
Multidisciplinary design optimization Multi-disciplinary design optimization (MDO) is a field of engineering that uses optimization methods to solve design problems incorporating a number of disciplines. It is also known as multidisciplinary system design optimization (MSDO), and Multi ...
*
Multiphysics In computational modelling, multiphysics simulation (often shortened to simply "multiphysics") is defined as the simultaneous simulation of different aspects of a physical system or systems and the interactions among them. For example, simultaneous ...
* Patch test *
Rayleigh–Ritz method The Rayleigh–Ritz method is a direct numerical method of approximating eigenvalues, originated in the context of solving physical boundary value problems and named after Lord Rayleigh and Walther Ritz. The name Rayleigh–Ritz is being debated ...
*
Space mapping The space mapping methodology for modeling and design optimization of engineering systems was first discovered by John Bandler in 1993. It uses relevant existing knowledge to speed up model generation and design optimization of a system. The kno ...
* STRAND7 *
Tessellation (computer graphics) In computer graphics, tessellation refers to the dividing of datasets of polygons (sometimes called ''vertex sets'') presenting objects in a scene into suitable structures for rendering. Especially for real-time rendering, data is tessellated ...
* Weakened weak form


References


Further reading

*G. Allaire and A. Craig:
Numerical Analysis and Optimization: An Introduction to Mathematical Modelling and Numerical Simulation
'. *K. J. Bathe: ''Numerical methods in finite element analysis'', Prentice-Hall (1976). *Thomas J.R. Hughes: ''The Finite Element Method: Linear Static and Dynamic Finite Element Analysis,'' Prentice-Hall (1987). *J. Chaskalovic: ''Finite Elements Methods for Engineering Sciences'', Springer Verlag, (2008). *
Endre Süli Endre Süli (also, Endre Suli or Endre Šili) is a mathematician. He is Professor of Numerical Analysis in the Mathematical Institute, University of Oxford, Fellow and Tutor in Mathematics at Worcester College, Oxford and Adjunct Fellow of Li ...

''Finite Element Methods for Partial Differential Equations''
*O. C. Zienkiewicz, R. L. Taylor, J. Z. Zhu :
The Finite Element Method: Its Basis and Fundamentals
', Butterworth-Heinemann (2005). *N. Ottosen, H. Petersson : ''Introduction to the Finite Element Method, '' Prentice-Hall (1992). *Zohdi, T. I. (2018) A finite element primer for beginners-extended version including sample tests and projects. Second Edition https://link.springer.com/book/10.1007/978-3-319-70428-9 {{Authority control Continuum mechanics Numerical differential equations Partial differential equations Structural analysis Computational electromagnetics Canadian inventions Russian inventions