The patch test in the
finite element method
The 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 ...
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 imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function.
The function is often thought of as an "unknown" to be sol ...
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 (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to objects r ...
, the prescribed exact solution consists of displacements that vary as piecewise linear functions in space (called a constant
strain
Strain may refer to:
Science and technology
* Strain (biology), variants of plants, viruses or bacteria; or an inbred animal used for experimental purposes
* Strain (chemistry), a chemical stress of a molecule
* Strain (injury), an injury to a mu ...
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 imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function.
The function is often thought of as an "unknown" to be sol ...
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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