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In this paper, we employ the linear virtual element spaces to discretize thesemilinear sine-Gordon equation in two dimensions. The salient features of thevirtual element method (VEM) are: (a) it does not require explicit form of theshape functions to construct the nonlinear and the bilinear terms, and (b)relaxes the constraint on the mesh topology by allowing the domain to bediscretized with general polygons consisting of both convex and concaveelements, and (c) easy mesh refinements (hanging nodes and interfaces areallowed). The nonlinear source term is discretized by employing the productapproximation technique and for temporal discretization, the Crank-Nicolsonscheme is used. The resulting nonlinear equations are solved using the Newton'smethod. We derive a priori error estimations in $L^2$ and $H^1$ norms. Theconvergence properties and the accuracy of the virtual element method for thesolution of the sine-Gordon equation are demonstrated with academic numericalexperiments.

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