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The development of new algorithms and computer codes for the solution of partial differential equations (PDEs) usually involves the use of proof-of-concept test problems. Such test problems have a variety of uses such as demonstrating that a new algorithm is effective, verifying that a new code is correct in the sense of achieving the theoretical order of convergence, and comparing the performance of different algorithms and codes.

Self-adaptive methods to determine a quasi-optimal grid are a critical component of the improvements that have been made in PDE algorithms in recent years. This field is referred to as adaptive mesh refinement, or adaptive grid refinement. Although adaptive mesh refinement techniques are now in widespread use in applications, they remain an active field of research, particularly in the context of hp-adaptive techniques, non-elliptic problems, accurate error estimators, etc.

Nearly every paper on algorithms for solving PDEs contains a numerical results section with one or more test problems. Although there are a few commonly used problems for adaptive mesh refinement research, such as the L-domain problem, the test problems vary widely among papers. The purpose of this web resource is to provide a standard set of problems suitable for benchmarking and testing adaptive mesh refinement algorithms and error estimators. Most of the problems are taken from the numerical results section of papers in the adaptive mesh refinement literature. The problems exhibit a variety of types of singularities (e.g. point and line singularities on the boundary and in the interior), near singularities (e.g. sharp peaks, boundary layers, and wave fronts), and other difficulties. Most of the problems are parameterized to allow "easy" and "hard" variations on the problem.

The collection of problems can be used for:

  • testing the correctness of a computer code
  • checking that an algorithm achieves the optimal rate of convergence
  • testing the effectivity index of error estimators
  • benchmarking the computation time of different computer codes and algorithms
  • other computations related to adaptive mesh refinement

To find the benchmark problems of interest to you, you define a filter to select the properties you want the problems to satisfy, such as 2D or 3D, or elliptic, parabolic or hyperbolic. A list of problems that satisfy those properties is presented, and you can select a problem to view from that list.

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Direct comments and questions to William Mitchell, Applied and Computational Mathematics Division, NIST

NOTE: This web site is under construction. Many categories do not have any entries yet. More test problems will be added, and existing test problems may be changed or removed. Please help by sending me any corrections or suggestions.

The National Institute of Standards and Technology (NIST) is an agency of the U.S. Department of Commerce.

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Date created: May 16, 2013 | Last updated: August 20, 2015    Contact: William Mitchell, Home Page
Development status: Active Development