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Domain Decomposition Methods
In mathematics, numerical analysis, and numerical partial differential equations, domain decomposition methods solve a boundary value problem by splitting it into smaller boundary value problems on subdomains and iterating to coordinate the solution between adjacent subdomains. A coarse problem with one or few unknowns per subdomain is used to further coordinate the solution between the subdomains globally. The problems on the subdomains are independent, which makes domain decomposition methods suitable for parallel computing. Domain decomposition methods are typically used as preconditioners for Krylov space iterative methods, such as the conjugate gradient method, GMRES, and LOBPCG. In overlapping domain decomposition methods, the subdomains overlap by more than the interface. Overlapping domain decomposition methods include the Schwarz alternating method and the additive Schwarz method. Many domain decomposition methods can be written and analyzed as a special case of the abst ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schwarz Alternating Method
In mathematics, the Schwarz alternating method or alternating process is an iterative method introduced in 1869–1870 by Hermann Schwarz in the theory of conformal mapping. Given two overlapping regions in the complex plane in each of which the Dirichlet problem could be solved, Schwarz described an iterative method for solving the Dirichlet problem in their union, provided their intersection was suitably well behaved. This was one of several constructive techniques of conformal mapping developed by Schwarz as a contribution to the problem of uniformization, posed by Riemann in the 1850s and first resolved rigorously by Koebe and Poincaré in 1907. It furnished a scheme for uniformizing the union of two regions knowing how to uniformize each of them separately, provided their intersection was topologically a disk or an annulus. From 1870 onwards Carl Neumann also contributed to this theory. In the 1950s Schwarz's method was generalized in the theory of partial differential ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mortar Method
In numerical analysis, mortar methods are discretization methods for partial differential equations, which use separate finite element discretization on nonoverlapping subdomains. The meshes on the subdomains do not match on the interface, and the equality of the solution is enforced by Lagrange multipliers, judiciously chosen to preserve the accuracy of the solution.Y. Maday, C. Mavriplis, and A. T. Patera, ''Nonconforming mortar element methods: application to spectral discretizations'', in Domain decomposition methods (Los Angeles, CA, 1988), SIAM, Philadelphia, PA, 1989, pp. 392--418. B. I. Wohlmuth, ''A mortar finite element method using dual spaces for the Lagrange multiplier'', SIAM J. Numer. Anal., 38 (2000), pp. 989--1012. Mortar discretizations lend themselves naturally to the solution by iterative domain decomposition methods such as FETI and balancing domain decomposition In numerical analysis, the balancing domain decomposition method (BDD) is an iterative method to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
FETI-DP
The FETI-DP method is a domain decomposition methodC. Farhat, M. Lesoinne, P. LeTallec, K. Pierson, and D. Rixen, ''FETI-DP: a dual-primal unified FETI method. I. A faster alternative to the two-level FETI method'', Internat. J. Numer. Methods Engrg., 50 (2001), pp. 1523--1544. that enforces equality of the solution at subdomain interfaces by Lagrange multipliers except at subdomain corners, which remain primal variables. The first mathematical analysis of the method was provided by Mandel and Tezaur.J. Mandel and R. Tezaur, ''On the convergence of a dual-primal substructuring method'', Numerische Mathematik, 88 (2001), pp. 543--558. The method was further improved by enforcing the equality of averages across the edges or faces on subdomain interfacesC. Farhat, M. Lesoinne, and K. Pierson, ''A scalable dual-primal domain decomposition method'', Numer. Linear Algebra Appl., 7 (2000), pp. 687--714. Preconditioning techniques for large sparse matrix problems in industrial applications ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lagrange Multiplier
In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function (mathematics), function subject to constraint (mathematics), equation constraints (i.e., subject to the condition that one or more equations have to be satisfied exactly by the chosen values of the variable (mathematics), variables). It is named after the mathematician Joseph-Louis Lagrange. Summary and rationale The basic idea is to convert a constrained problem into a form such that the derivative test of an unconstrained problem can still be applied. The relationship between the gradient of the function and gradients of the constraints rather naturally leads to a reformulation of the original problem, known as the Lagrangian function or Lagrangian. In the general case, the Lagrangian is defined as \mathcal(x, \lambda) \equiv f(x) + \langle \lambda, g(x)\rangle for functions f, g; the notation \langle \cdot, \cdot \rangle denotes an inner prod ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
BDDC
In numerical analysis, BDDC (balancing domain decomposition by constraints) is a domain decomposition method for solving large symmetric matrix, symmetric, positive definite matrix, positive definite systems of linear equations that arise from the finite element method. BDDC is used as a preconditioner to the conjugate gradient method. A specific version of BDDC is characterized by the choice of coarse degrees of freedom, which can be values at the corners of the subdomains, or averages over the edges or the faces of the interface between the subdomains. One application of the BDDC preconditioner then combines the solution of local problems on each subdomains with the solution of a global coarse problem with the coarse degrees of freedom as the unknowns. The local problems on different subdomains are completely independent of each other, so the method is suitable for parallel computing. With a proper choice of the coarse degrees of freedom (corners in 2D, corners plus edges or corner ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Balancing Domain Decomposition
In numerical analysis, the balancing domain decomposition method (BDD) is an iterative method to find the solution of a symmetric positive definite system of linear algebraic equations arising from the finite element method.J. Mandel, ''Balancing domain decomposition'', Comm. Numer. Methods Engrg., 9 (1993), pp. 233–241. In each iteration, it combines the solution of local problems on non-overlapping subdomains with a coarse problem created from the subdomain nullspaces. BDD requires only solution of subdomain problems rather than access to the matrices of those problems, so it is applicable to situations where only the solution operators are available, such as in oil reservoir simulation by mixed finite elements.L. C. Cowsar, J. Mandel, and M. F. Wheeler, ''Balancing domain decomposition for mixed finite elements'', Math. Comp., 64 (1995), pp. 989–1015. In its original formulation, BDD performs well only for 2nd order problems, such elasticity in 2D and 3D. Fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abstract Additive Schwarz Method
In mathematics, the abstract additive Schwarz method, named after Hermann Schwarz, is an abstract version of the additive Schwarz method for boundary value problems on partial differential equations, formulated only in terms of linear algebra Linear algebra is the branch of mathematics concerning linear equations such as :a_1x_1+\cdots +a_nx_n=b, linear maps such as :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrix (mathemat ... without reference to domains, subdomains, etc. Many if not all domain decomposition methods can be cast as abstract additive Schwarz method, which is often the first and most convenient approach to their analysis.. References Domain decomposition methods {{mathapplied-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Additive Schwarz Method
In mathematics, the additive Schwarz method, named after Hermann Schwarz, solves a boundary value problem for a partial differential equation approximately by splitting it into boundary value problems on smaller domains and adding the results. Overview Partial differential equations (PDEs) are used in all sciences to model phenomena. For the purpose of exposition, we give an example physical problem and the accompanying boundary value problem (BVP). Even if the reader is unfamiliar with the notation, the purpose is merely to show what a BVP looks like when written down. :(Model problem) The heat distribution in a square metal plate such that the left edge is kept at 1 degree, and the other edges are kept at 0 degree, after letting it sit for a long period of time satisfies the following boundary value problem: ::''f''''xx''(''x'',''y'') + ''f''''yy''(''x'',''y'') = 0 ::''f''(0,''y'') = 1; ''f''(''x'',0) = ''f''(''x'',1) = ''f''(1,''y'') = 0 :where ''f'' is the unknown functi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
LOBPCG
Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) is a matrix-free method for finding the largest (or smallest) eigenvalues and the corresponding eigenvectors of a symmetric generalized eigenvalue problem :A x= \lambda B x, for a given pair (A, B) of complex Hermitian or real symmetric matrices, where the matrix B is also assumed positive-definite. Background Kantorovich in 1948 proposed calculating the smallest eigenvalue \lambda_1 of a symmetric matrix A by steepest descent using a direction r = Ax-\lambda (x) x of a scaled gradient of a Rayleigh quotient \lambda(x) = (x, Ax)/(x, x) in a scalar product (x, y) = x'y, with the step size computed by minimizing the Rayleigh quotient in the linear span of the vectors x and r, i.e. in a locally optimal manner. Samokish proposed applying a preconditioner T to the residual vector r to generate the preconditioned direction w = T r and derived asymptotic, as x approaches the eigenvector, convergence rate bounds. D'yakonov ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
