Commun. Comput. Phys.,
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Volume 3.

Enforcing the Discrete Maximum Principle for Linear Finite Element Solutions of Second-Order Elliptic Problems

Richard Liska 1*, Mikhail Shashkov 2

1 Czech Technical University in Prague, Faculty of Nuclear Sciences and Physical Engineering, Brehova 7, 115 19 Prague 1, Czech Republic.
2 Theoretical Division, Group T-7, MS-B284, Los Alamos National Laboratory, Los Alamos, NM 87545, USA.

Received 29 June 2007; Accepted (in revised version) 18 September 2007
Available online 11 December 2007


The maximum principle is a basic qualitative property of the solution of second-order elliptic boundary value problems. The preservation of the qualitative characteristics, such as the maximum principle, in discrete model is one of the key requirements. It is well known that standard linear finite element solution does not satisfy maximum principle on general triangular meshes in 2D. In this paper we consider how to enforce discrete maximum principle for linear finite element solutions for the linear second-order self-adjoint elliptic equation. First approach is based on repair technique, which is a posteriori correction of the discrete solution. Second method is based on constrained optimization. Numerical tests that include anisotropic cases demonstrate how our method works for problems for which the standard finite element methods produce numerical solutions that violate the discrete maximum principle.

AMS subject classifications: 35J25, 65N99

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Key words: Second-order elliptic problems, linear finite element solutions, discrete maximum principle, constrained optimization.

*Corresponding author.
Email: (R. Liska), (M. Shashkov)

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