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Non-conforming FEM for the quasi-static contact problem

Kamana Porwal, Tanvi Wadhawan

Abstract

In this article, we addressed the numerical solution of a non-linear evolutionary variational inequality, which is encountered in the investigation of quasi-static contact problems. Our study encompasses both the semi-discrete and fully-discrete schemes, where we employ the backward Euler method for time discretization and utilize the lowest order Crouzeix-Raviart nonconforming finite element method for spatial discretization. By assuming appropriate regularity conditions on the solution, we establish \emph{a priori} error analysis for these schemes, achieving the optimal convergence order for linear elements. To illustrate the numerical convergence rates, we provide numerical results on a two-dimensional test problem.

Non-conforming FEM for the quasi-static contact problem

Abstract

In this article, we addressed the numerical solution of a non-linear evolutionary variational inequality, which is encountered in the investigation of quasi-static contact problems. Our study encompasses both the semi-discrete and fully-discrete schemes, where we employ the backward Euler method for time discretization and utilize the lowest order Crouzeix-Raviart nonconforming finite element method for spatial discretization. By assuming appropriate regularity conditions on the solution, we establish \emph{a priori} error analysis for these schemes, achieving the optimal convergence order for linear elements. To illustrate the numerical convergence rates, we provide numerical results on a two-dimensional test problem.
Paper Structure (7 sections, 3 theorems, 99 equations, 1 figure, 2 tables)

This paper contains 7 sections, 3 theorems, 99 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Discrete Problem
  3. Semi-Discrete Approximation
  4. Error Analysis for Spatially Semi-Discrete Approximation
  5. Fully-Discrete Approximation
  6. Error Estimates for Fully-Discrete Approximation
  7. Numerical Experiments

Key Result

Lemma 2.3

For any $\boldsymbol{v} \in \boldsymbol{H^\nu}(T), T \in \mathcal{T}_h$, the following holds where $~1\leq\nu\leq2$.

Figures (1)

  • Figure 5.1: Physical Setting for Example 5.1.

Theorems & Definitions (5)

  • Lemma 2.3
  • Theorem 3.1
  • proof
  • Theorem 4.1
  • proof