Discrete maximum principles for finite element solutions of nonlinear elliptic problems with mixed boundary conditions

Abstract

One of the most important problems in numerical simulations is the preservation of qualitative properties of solutions of the mathematical models by computed approximations. For problems of elliptic type, one of the basic properties is the (continuous) maximum principle. In our work, we present several variants of the maximum principles and their discrete counterparts for (scalar) second-order nonlinear elliptic problems with mixed boundary conditions. The problems considered are numerically solved by the continuous piecewise linear finite element approximations built on simplicial meshes. Sufficient conditions providing the validity of the corresponding discrete maximum principles are presented. Geometrically, they mean that the employed meshes have to be of acute or nonobtuse type, depending of the type of the problem. Finally some examples of real-life problems, where the preservation of maximum principles plays an important role, are presented.

Discrete maximum principle for FE solutions of the diffusion-reaction problem on prismatic meshes

“…The following theorem shows the continuous maximum principle (CMP) for problem (1), see [24] and also [17,18] for a more general case of nonlinear problems with mixed boundary conditions. In what follows, the equalities and inequalities between functions from Lebesgue spaces should be understood up to a set of zero measure, as usual.…”

“…[6,8,17,18,21,25,28,29]. They discuss various numerical methods for different problems and study the validity of the DMPs.…”

“…Among these, the maximum principle is the basic characteristic usually associated with the second order elliptic (and parabolic) boundary value problems [17,24,25]. It can be mathematically described as an a priori estimate of the magnitude of the solution (unknown in the whole domain) by the magnitude of the given (i.e.…”

Finite element approximation to the extremal eigenvalue problem for inhomogenous materials

“…As proved in Theorem 4 of [18] or Theorem 3 of [22], if matrix M h is irreducibly diagonally dominant, and (2) of Assumption 3.3 is fulfilled, we have (M h ) −1 is a positive matrix and hence discrete strong maximum principle holds (i.e.,…”

There Is No Face-to-Face Partition of R5 into Acute Simplices

“…Note that acute simplicial partitions (defined in Section 2 below) are very useful in numerical analysis, since they yield irreducible and diagonally dominant stiffness matrices, when solving the equation − u + bu = f by standard linear conforming finite elements in a bounded polytopic domain in R d with some boundary conditions and b ≥ 0 small enough. In this case the discrete maximum principle takes place, see [4] (and also [5] for nonlinear problems). The necessity of solving partial differential equations for dimensions d > 3 arises in statistical physics, financial mathematics, general relativity, particle physics, etc.…”