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Sufficient conditions for strong discrete maximum principles in finite element solutions of linear and semilinear elliptic equations

Andrei Draganescu, L. Ridgway Scott

Abstract

We introduce a novel technique for proving global strong discrete maximum principles for finite element discretizations of linear and semilinear elliptic equations for cases when the common, matrix-based sufficient conditions are not satisfied. The basic argument consists of extending the strong form of discrete maximum principle from macroelements to the entire domain via a connectivity argument. The method is applied to discretizations of elliptic equations with certain pathological meshes, and to semilinear elliptic equations.

Sufficient conditions for strong discrete maximum principles in finite element solutions of linear and semilinear elliptic equations

Abstract

We introduce a novel technique for proving global strong discrete maximum principles for finite element discretizations of linear and semilinear elliptic equations for cases when the common, matrix-based sufficient conditions are not satisfied. The basic argument consists of extending the strong form of discrete maximum principle from macroelements to the entire domain via a connectivity argument. The method is applied to discretizations of elliptic equations with certain pathological meshes, and to semilinear elliptic equations.

Paper Structure

This paper contains 23 sections, 20 theorems, 157 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Motivation and problem formulation
  3. The continuous model problems and their maximum principles
  4. Finite element discretization
  5. Abstract strong discrete maximum principles
  6. Abstract solution operators
  7. Main results
  8. Matrix form of sDMP-A and sDMP-B for linear elliptic equations
  9. Sufficient conditions for a weak discrete maximum principle
  10. Application to linear elliptic equations
  11. Connection to the classical results
  12. The angle condition
  13. Defects related to boundary mesh properties
  14. Defects related to interior mesh properties
  15. A class of meshes with defects
  16. ...and 8 more sections

Key Result

Theorem 2.1

Assume $c\equiv 0$ in eq:slcontdef, $a_{ij}$ are continuously differentiable, and $u\in C^2(D) \cap C^1(\overline{D})$ solves eq:slcontdef-eq:slcontBC. (i) If $f\ge 0$, then In addition, if $u$ attains a minimum over $\overline{D}$ at $x_0\in D$ (i.e., an interior point), then $u$ is constant. (ii) If $f\le 0$, then In addition, if $u$ attains a maximum over $\overline{D}$ at $x_0\in D$ (i.e.,

Figures (13)

  • Figure 1: Triangle elements entering the formulas for stiffness and mass matrices.
  • Figure 2: Left: The Poisson solution operator $S^D_h$ satisfies wDMP-A on the classical three-line mesh on $D = [0,1]\times [0,1]$ (here $n=5$), but does not satisfy sDMP-A; entries in the stiffness matrix corresponding to edges like $\overline{P_9, P_{16}}$ are zero. Right: If $\tilde{D}$ is obtained by removing from $D$ the triangles containing $P_6$ and $P_{31}$, then sDMP-A holds on $\tilde{D}$, and wDMP-A holds on ${D}$.
  • Figure 3: The mesh $G_1(\pi/3)$ contains 1 edge violating the angle condition, but ${\mathbf A}^{-1}_1 >{\mathbf 0}$ (verified numerically -- see also Fig. \ref{['fig:minGreensfunctions']}).
  • Figure 4: The mesh $G_2(2\pi/5)$ contains 4 edges violating the angle condition, but ${\mathbf A}^{-1}_2 >{\mathbf 0}$ (verified numerically -- see also Fig. \ref{['fig:minGreensfunctions']}).
  • Figure 5: The smallest values of ${\mathbf A}^{-1}_1(\theta),\dots,{\mathbf A}^{-1}_4(\theta)$ are shown as functions of $\theta$ (on the $x$-axis). Note that each of the four functions has a root $\theta_k <\pi/2$ so that it is positive for $\theta_k< \theta < \pi/2$.
  • ...and 8 more figures

Theorems & Definitions (45)

  • Theorem 2.1
  • Theorem 2.2
  • Lemma 2.3
  • proof
  • Definition 3.1
  • Definition 3.2
  • Remark 3.3
  • Theorem 3.4
  • proof
  • Corollary 3.5
  • ...and 35 more