Enforcing the Discrete Maximum Principle for Linear Finite Element Solutions of Second-Order Elliptic Problems (Communications in Computational Physics, 2008)
The maximum principle is a basic qualitative property of the solution ofsecond-order elliptic boundary value problems. The preservation of the qualitativecharacteristics, such as the maximum principle, in discrete model is one of the keyrequirements. It is well known that standard linear finite element solution does notsatisfy maximum principle on general triangular meshes in 2D. In this paper we consider how to enforce discrete maximum principle for linear finite element solutions forthe linear second-order self-adjoint elliptic equation. First approach is based on repairtechnique, which is a posteriori correction of the discrete solution. Second methodis based on constrained optimization. Numerical tests that include anisotropic casesdemonstrate how our method works for problems for which the standard finite element methods produce numerical solutions that violate the discrete maximum principle.