Sufficient Conditions for Strong Discrete Maximum Principles in Finite Element Solutions of Linear and Semilinear Elliptic Equations

Jul 16, 2026

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A novel technique establishes global strong discrete maximum principles for finite element discretizations of linear and semilinear elliptic equations in situations where common matrix-based sufficient conditions fail. The approach globalizes the strong form from macroelements to the full domain using a connectivity argument, rather than requiring nonpositive off-diagonal matrix entries everywhere. Applications cover discretizations on pathological meshes and semilinear elliptic problems, extending the scope beyond standard M-matrix frameworks and clarifying what guarantees global sDMP.

LanguageEnglish

CategoryResearch & Report

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Pages40

Reading Time1 h 41 min

Introduction

Motivation and discrete maximum principles

Motivation and problem formulation

Continuous model problems and maximum principles

Organization of the paper

Outline of main results

arXiv :2509 .00932v3 [math .NA] 13 Jul 2026 MATHEMATICS OF COMPUTATION Volume 00, Number 0, Pages 000–000 S 0025-5718(XX)0000-0 SUFFICIENT CONDITIONS FOR STRONG DISCRETE MAXIMUM PRINCIPLES IN FINITE ELEMENT SOLUTIONS OF LINEAR AND SEMILINEAR ELLIPTIC EQUATIONS ANDREI DR˘AG˘ANESCU AND 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. 1. Introduction The preservation of qualitative properties constitutes a central theme in the design of numerical methods for partial differential equations. Among those properties, maximum principles have captured the attention of many generations of numerical analysts, as they play an essential role in ensuring that solutions maintain their physical relevance. For example, it is not only desired, but sometimes critical that quantities representing concentrations lie in the interval [0,1], that computed densities are positive, or that fluxes across interfaces have the correct sign. In this work we focus on discrete maximum principles (DMPs) for finite element solutions of linear and semilinear elliptic equations. A short and particular formulation of the DMP is that the maximum of a discrete subharmonic function cannot be achieved in the interior of its domain unless the function is constant. At the discrete level we distinguish between the local DMP, which refers to the DMP being satisfied on the union of elements with a common vertex, and the global DMP, which refers to the entire domain; verifying the latter is the ultimate goal, but also presents a greater challenge. A long list of works [7, 27, 33, 21, 16, 17, 19, 31, 30, 13, 32, 18, 1](to cite only a few) was devoted to studying conditions under which appropriate forms of the global DMP hold for linear and nonlinear elliptic equations. Many more references can be found in the review article [2] and the recent monograph [3] . A common element for all these articles is the hypothesis that certain key matrices have nonpositive off-diagonal elements; for the case of the linear Poisson equation, this reduces to the necessity for the stiffness matrix to be an M-matrix (see [29] for definition) . Not only is this condition restrictive on the mesh (see Definition 2020 Mathematics Subject Classification. Primary 65N30 . The first author was supported in part by NSF Award 2409951 . 2 ANDREI DR˘AG˘ANESCU AND L. RIDGWAY SCOTT 2.2 in [2]), but it is equivalent to the local DMP to be satisfied around each vertex. Hence, it is fair to say that most of the aforementioned works use various techniques to globalize the DMP, after essentially assuming the local DMP holds everywhere. However, in [8] (Section 6) it is shown that the global DMP can hold on certain meshes where local DMPs do not hold. Therefore, the nonpositivity of the off-diagonal entries is not a necessary condition for the global DMP (see also [3], Example 6.10, p. 159) . Moreover, an example is given [8] where the global DMP fails, even as the mesh size converges to zero. It is notable how challenging it is to find an example where the finite element spaces have good approximation properties, but the global DMP fails; in fact the global DMP seems to hold for many practical situations. Hence, it is fair to say that there is a gap in the literature between the known sufficient and the necessary conditions for the global DMP to hold. In this paper we are providing a set of conditions that aim to bridge this gap. We also note that approximation alone allows proving a wea

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What is the main method used to obtain the global strong discrete maximum principle?

The key step extends the strong discrete maximum principle from macroelements to the entire domain via a connectivity argument, without assuming nonpositive off-diagonal entries everywhere in the stiffness matrix.

Jul 16, 2026

Upload by Himbo