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

Hongliang Li, Pingbing Ming, Yinghong Zhou

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.

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

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.
Paper Structure (7 sections, 11 theorems, 102 equations, 1 figure, 3 tables)

This paper contains 7 sections, 11 theorems, 102 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. TRUNC finite element space
  3. TRUNC-type finite element approximation
  4. Numerical Experiments
  5. Example of smooth solution
  6. Example of solution with boundary layer
  7. Conclusion

Key Result

Lemma 2.1

For the TRUNC-type element, the $\Sigma_K$ is $Z_K$-unisolvent. Moreover, let $\phi_i$ and $\phi_{ij}$ be the basis functions associated with the degree freedom $p(a_i)$ and $e_{ij}\cdot\nabla p(a_i)$, respectively. Then

Figures (1)

  • Figure 1: Plots of meshes.

Theorems & Definitions (19)

  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • Theorem 3.1
  • Lemma 3.2
  • Corollary 3.3
  • ...and 9 more