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

Discrete minimum and maximum principles for finite element approximations of non-monotone elliptic equations

Uniform lower and upper bounds for positive finite-element approximations to semilinear elliptic equations in several space dimensions subject to mixed Dirichlet-Neumann boundary conditions are derived. The main feature is that the non-linearity may be non-monotone and unbounded. The discrete minimum principle provides a positivity-preserving approximation if the discretization parameter is sma...

Mortar Finite Volume Element Approximations of Second Order Elliptic Problems

New Maximum Principles for Linear Elliptic Equations

We prove extensions of the estimates of Aleksandrov and Bakel′man for linear elliptic operators in Euclidean space R to inhomogeneous terms in L spaces for q < n. Our estimates depend on restrictions on the ellipticity of the operators determined by certain subcones of the positive cone. We also consider some applications to local pointwise and L estimates.

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

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 pa...

A Posteriori Error Estimates for Mixed Finite Element Galerkin Approximations to Second Order Linear Hyperbolic Equations

In this article, a posteriori error analysis for mixed finite element Galerkin approximations of second order linear hyperbolic equations is discussed. Based on mixed elliptic reconstructions and an integration tool, which is a variation of Baker’s technique introduced earlier by G. Baker (SIAM J. Numer. Anal., 13 (1976), 564-576) in the context of a priori estimates for a second order wave equ...