Virtual element method for semilinear sine-Gordon equation over polygonal mesh using product approximation technique

TL;DR

This paper introduces a virtual element method for solving the semilinear sine-Gordon equation on polygonal meshes, utilizing product approximation and Crank-Nicolson schemes, with proven convergence and accuracy through numerical experiments.

Contribution

It presents a novel application of the virtual element method to the sine-Gordon equation, allowing flexible polygonal meshes and avoiding explicit shape functions.

Findings

Demonstrates convergence in $L^2$ and $H^1$ norms.

Shows the method's effectiveness on general polygonal meshes.

Provides error estimates and numerical validation.

Abstract

In this paper, we employ the linear virtual element spaces to discretize the semilinear sine-Gordon equation in two dimensions. The salient features of the virtual element method (VEM) are: (a) it does not require explicit form of the shape functions to construct the nonlinear and the bilinear terms, and (b) relaxes the constraint on the mesh topology by allowing the domain to be discretized with general polygons consisting of both convex and concave elements, and (c) easy mesh refinements (hanging nodes and interfaces are allowed). The nonlinear source term is discretized by employing the product approximation technique and for temporal discretization, the Crank-Nicolson scheme is used. The resulting nonlinear equations are solved using the Newton's method. We derive a priori error estimations in $L^2$ and $H^1$ norms. The convergence properties and the accuracy of the virtual element method for the solution of the sine-Gordon equation are demonstrated with academic numerical experiments.