The TRUNC element in any dimension and application to a modified Poisson equation

TL;DR

This paper introduces a new finite element called TRUNC in multiple dimensions, applies it to a modified Poisson equation from electrostatics, and validates its effectiveness through theoretical error estimates and numerical tests.

Contribution

It presents the first multi-dimensional TRUNC finite element and demonstrates its application to a complex electrostatics problem with rigorous error analysis.

Findings

Uniform error estimate established

Numerical tests confirm theoretical predictions

Effective approximation of solutions with boundary layers

Abstract

We introduce a novel TRUNC finite element in n dimensions, encompassing the traditional TRUNC triangle as a particular instance. By establishing the weak continuity identity, we identify it as crucial for error estimate. This element is utilized to approximate a modified Poisson equation defined on a convex polytope, originating from the nonlocal electrostatics model. We have substantiated a uniform error estimate and conducted numerical tests on both the smooth solution and the solution with a sharp boundary layer, which align with the theoretical predictions.