Patch test (finite elements)

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The patch test in the finite element method is a simple indicator of the quality of a finite element, developed by Bruce Irons. The patch test uses a partial differential equation on a domain consisting from several elements set up so that the exact solution is known. Typically, in mechanics, the prescribed exact solution consists of displacements that vary as 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 as the finite element mesh is refined. However, this is not the case, and the patch test is neither sufficient nor necessary for convergence.