Discrete maximum principle for a 1D problem with piecewise-constant coefficients solved by hp-FEM

Abstract

In this paper we prove the discrete maximum principle for a one-dimensional equation of the form –(au′)′ = f with piecewise-constant coefficient a(x), discretized by the hp-FEM. The discrete problem is transformed in such a way that the discontinuity of the coefficient a(x) disappears. Existing results are then applied to obtain a condition on the mesh which guarantees the satisfaction of the discrete maximum principle. Both Dirichlet and mixed Dirichlet–Neumann boundary conditions are discussed.