Virtual element method for semilinear sineGordon equation over polygonal mesh using product approximation technique Abstract   In this paper, we employ the linear virtual element spaces to discretize the semilinear sinexe2x80x93Gordon 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 Crankxe2x80x93Nicolson scheme is used. The resulting nonlinear equations are solved using the Newtonxe2x80x99s 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 sinexe2x80x93Gordon equation are demonstrated with academic numerical experiments.

Superconvergent Functional Estimates from TensorProduct Generalized SummationbyParts Discretizations in Curvilinear Coordinates We investigate superconvergent functional estimates in curvilinear coordinates for diagonalnorm tensorproduct generalized summationbyparts operators. We show that interpolationextrapolation operators of degree greater than or equal to 2p are required to preserve at least 2p quadrature accuracy and functional superconvergence in curvilinear coordinates when: (1) the Jacobian of the coordinate transformation is approximated by the same generalized summationbyparts operator that is used to approximate the flux terms and (2) the degree of the generalized summationbyparts operator is lower than the degree of the polynomial used to represent the geometry of interest. Legendrexe2x80x93Gaussxe2x80x93Lobatto and Legendrexe2x80x93Gauss elementtype operators are considered. When the aforementioned condition (2) is violated for the Legendrexe2x80x93Gauss operators, there is an evenxe2x80x93odd quadrature convergence pattern that is explained by the cancellation of the leading truncation error terms for the interpolationextrapolation operators that correspond to the odddegree Legendrexe2x80x93Gauss operators. The theory developed is confirmed through numerical examples with a steady onedimensional problem and the unsteady twodimensional linear convection equation.