The patch test in the
finite element method
Finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat tran ...
is a simple indicator of the quality of a finite element, developed by
Bruce Irons.
The patch test uses a
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives.
The function is often thought of as an "unknown" that solves the equation, similar to ho ...
on a domain consisting from several elements set up so that the exact solution is known and can be reproduced, in principle, with zero error. Typically, in
mechanics
Mechanics () is the area of physics concerned with the relationships between force, matter, and motion among Physical object, physical objects. Forces applied to objects may result in Displacement (vector), displacements, which are changes of ...
, the prescribed exact solution consists of displacements that vary as piecewise linear functions in space (called a constant
strain solution). The elements pass the patch test if the finite element solution is the same as the exact solution.
It was long conjectured by engineers that passing the patch test is sufficient for the convergence of the finite element, that is, to ensure that the solutions from the finite element method converge to the exact solution of the
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives.
The function is often thought of as an "unknown" that solves the equation, similar to ho ...
as the finite element mesh is refined. However, this is not the case, and the patch test is neither sufficient nor necessary for convergence.
A broader definition of patch test (applicable to any numerical method, including and beyond finite elements) is any test problem having an exact solution that can, in principle, be exactly reproduced by the numerical approximation. Therefore, a finite-element simulation that uses linear shape functions has patch tests for which the exact solution must be piecewise linear, while higher-order finite elements have correspondingly higher-order patch tests.
References
Finite element method
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