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Explicit Consistency Error Estimate for Finite Element Solutions of the Poisson Equation on Convex Domains

Su Ruibo

TL;DR

This work addresses explicit a priori consistency error estimates for finite element solutions of the Poisson equation on convex domains when the domain is approximated by an internal convex polyhedron. The authors decompose the total error into boundary perturbation, domain/data approximation, and FEM discretization components, and derive bounds that depend only on global geometric parameters and standard seminorms. They provide dimension-specific, explicit constants for the interpolation error (notably $A_h$), with 2D bounds based on the maximal circumradius or mesh-geometry (e.g., minimal angle) and a 3D bound under standard mesh regularity. A boundary-perturbation analysis and source-term approximation are combined to yield a practical $L^2$ error bound, which is validated by a numerical example on the unit disk using polygonal domain approximations. The results enable mesh-agnostic, computable consistency estimates that are valuable for rigorous FEM error budgeting on convex domains.

Abstract

We derive explicit a priori consistency error estimates for a standard finite element discretization of the Poisson equation on convex domains, where the domain is approximated by an internal convex polyhedron. The obtained explicit estimates depend only on global geometric parameters and are applicable to general convex domains and arbitrary families of simplicial meshes.

Explicit Consistency Error Estimate for Finite Element Solutions of the Poisson Equation on Convex Domains

TL;DR

This work addresses explicit a priori consistency error estimates for finite element solutions of the Poisson equation on convex domains when the domain is approximated by an internal convex polyhedron. The authors decompose the total error into boundary perturbation, domain/data approximation, and FEM discretization components, and derive bounds that depend only on global geometric parameters and standard seminorms. They provide dimension-specific, explicit constants for the interpolation error (notably ), with 2D bounds based on the maximal circumradius or mesh-geometry (e.g., minimal angle) and a 3D bound under standard mesh regularity. A boundary-perturbation analysis and source-term approximation are combined to yield a practical error bound, which is validated by a numerical example on the unit disk using polygonal domain approximations. The results enable mesh-agnostic, computable consistency estimates that are valuable for rigorous FEM error budgeting on convex domains.

Abstract

We derive explicit a priori consistency error estimates for a standard finite element discretization of the Poisson equation on convex domains, where the domain is approximated by an internal convex polyhedron. The obtained explicit estimates depend only on global geometric parameters and are applicable to general convex domains and arbitrary families of simplicial meshes.

Paper Structure

This paper contains 16 sections, 5 theorems, 84 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Domain and Function Spaces
  4. Approximation of Domain
  5. Triangulation and Finite Element Space
  6. Finite Element Discretization
  7. Numerical Solution
  8. Upper Bound of Consistency Error
  9. Boundary Perturbation
  10. Approximation of Source term
  11. Total consistency Error
  12. Evaluation by maximal circumradius in two dimensions.
  13. Evaluation by mesh size and minimal angle in two dimensions.
  14. Evaluation by mesh size for non-blunt meshes in two dimensions
  15. Evaluation by regularity and mesh size in three dimensions
  16. ...and 1 more sections

Key Result

Theorem 3.1

Let $u$ be the weak solution to ExactProblem, $u_h$ be the finite element solution solution to FEM. Then Here, $C_P(\Omega_{\delta})$ denotes the Poincaré constant of $\Omega_{\delta}$, defined as the inverse of the square root of the smallest eigenvalue of and it satisfies the geometric bound Moreover, which depends explicitly on the space dimension $n$ and the global geometric properties of

Figures (4)

  • Figure 1: gap width
  • Figure 2: Hierarchical decomposition of the total consistency error.
  • Figure 3: $\alpha$,$\beta$,$\theta$ of triangle T
  • Figure 4: Constraints of $(a,b)$

Theorems & Definitions (16)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3: Local interpolation error constants
  • Theorem 3.1: Explicit Consistency Error
  • Remark 3.2: Upper Bound of $\|f-f_h\|_{L^2(\Omega_{\delta})}$
  • Corollary 3.3
  • Definition 4.1
  • Lemma 4.2
  • proof
  • Remark 4.3: Optimality
  • ...and 6 more