Maximum Principles for P1-Conforming Finite Element Approximations of Quasi-Linear Second Order Elliptic Equations

TL;DR

This paper extends classical maximum principles to $P1$-conforming finite element methods for quasi-linear second order elliptic equations, using variational approaches and angle conditions on mesh partitions.

Contribution

It establishes discrete maximum principles for finite element approximations of quasi-linear elliptic equations under specific mesh angle conditions, extending PDE theory to numerical methods.

Findings

Maximum principles hold under angle conditions on mesh elements.

Angle conditions involve $ ext{O}(h^eta)$-acuteness of triangles or tetrahedra.

Results apply to Poisson problems with non-obtuse or non-negative interior edges.

Abstract

This paper derives some discrete maximum principles for $P1$-conforming finite element approximations for quasi-linear second order elliptic equations. The results are extensions of the classical maximum principles in the theory of partial differential equations to finite element methods. The mathematical tools are based on the variational approach that was commonly used in the classical PDE theory. The discrete maximum principles are established by assuming a property on the discrete variational form that is of global nature. In particular, the assumption on the variational form is verified when the finite element partition satisfies some angle conditions. For the general quasi-linear elliptic equation, these angle conditions indicate that each triangle or tetrahedron needs to be $\mathcal{O}(h^\alpha)$-acute in the sense that each angle $\alpha_{ij}$ (for triangle) or interior dihedral angle $\alpha_{ij}$ (for tetrahedron) must satisfy $\alpha_{ij}\le \pi/2-\gamma h^\alpha$ for some $\alpha\ge 0$ and $\gamma>0$. For the Poisson problem where the differential operator is given by Laplacian, the angle requirement is the same as the existing ones: either all the triangles are non-obtuse or each interior edge is non-negative. It should be pointed out that the analytical tools used in this paper are based on the powerful De Giorgi's iterative method that has played important roles in the theory of partial differential equations. The mathematical analysis itself is of independent interest in the finite element analysis.