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A posteriori error analysis of the virtual element method for second-order quasilinear elliptic PDEs

Scott Congreve, Alice Hodson

Abstract

In this paper we develop a $C^0$-conforming virtual element method (VEM) for a class of second-order quasilinear elliptic PDEs in two dimensions. We present a posteriori error analysis for this problem and derive a residual based error estimator. The estimator is fully computable and we prove upper and lower bounds of the error estimator which are explicit in the local mesh size. We use the estimator to drive an adaptive mesh refinement algorithm. A handful of numerical test problems are carried out to study the performance of the proposed error indicator.

A posteriori error analysis of the virtual element method for second-order quasilinear elliptic PDEs

Abstract

In this paper we develop a -conforming virtual element method (VEM) for a class of second-order quasilinear elliptic PDEs in two dimensions. We present a posteriori error analysis for this problem and derive a residual based error estimator. The estimator is fully computable and we prove upper and lower bounds of the error estimator which are explicit in the local mesh size. We use the estimator to drive an adaptive mesh refinement algorithm. A handful of numerical test problems are carried out to study the performance of the proposed error indicator.
Paper Structure (13 sections, 10 theorems, 79 equations, 1 figure)

This paper contains 13 sections, 10 theorems, 79 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The virtual element method
  3. Mesh regularity and a polynomial approximation result
  4. The discrete spaces and projection operators
  5. Global spaces and the discrete forms
  6. The discrete problem
  7. A posteriori error analysis
  8. Approximation properties
  9. The residual equation
  10. Upper bounds (reliability)
  11. Lower bounds (efficiency)
  12. Numerical results
  13. Problem 1: smooth solution

Key Result

Theorem 2.3

Under Assumption ass: mesh regularity, for any $k \geq 0$ and for any $w \in H^m(E)$ with $1 \leq m \leq k+1$, it holds that where the constant $C_{3}$ depends only on $k$ and the mesh regularity.

Figures (1)

  • Figure :

Theorems & Definitions (22)

  • Remark 2.2
  • Theorem 2.3: Approximation using polynomials
  • Definition 2.4
  • Definition 2.6
  • Definition 2.7
  • Lemma 2.8
  • Definition 2.9: Stabilisation
  • Remark 2.10
  • Theorem 2.11: Existence and uniqueness of a discrete solution
  • Lemma 2.12
  • ...and 12 more