The finite element patch test revisited a computer test for convergence, validation and error estimates Original Research Article

he patch test was introduced in the mid-1960s, first as a check of satisfactory performance for elements violating continuity requirements, and later simply as a necessary condition which all elements had to satisfy and which frequently checked simple programming errors. In its original form the patch test simply verified that the minimum local consistency error was achievable and that the programming was correct. Later, it was established that the test could also be used to find the convergence order of any finite element type. While consistency is one of the necessary requirements for convergence the second is of course that of stability. In the original early work only displacement type, structural elements were involved and thus stability was ensured by rejecting elements which could form mechanisms. In a 1986 reformulation of the patch test, this stability (or solvability) requirement was formalized to all element assemblies and the patch test extended to become a generally applicable sufficient and necessary condition for convergence of all element forms [1,2]. Although some misinterpretation of the patch test has occurred in literature [3,4], the authors believe that with certain restrictions on regularity of the problem dealt with, the satisfaction of the patch test is equivalent to the satisfaction of the more mathematical Babuška-Brezzi criteria [5,6]. In recent years another important use of the patch test has been found by Babuška and his coworkers [7–9]. This work tests the asymptotic effectivity of various error estimating procedures used with various element patterns.