Time discontinuous Galerkin space-time finite element method for nonlinear Sobolev equations

Abstract

This article presents a complete discretization of a nonlinear Sobolev equation using space-time discontinuous Galerkin method that is discontinuous in time and continuous in space. The scheme is formulated by introducing the equivalent integral equation of the primal equation. The proposed scheme does not explicitly include the jump terms in time, which represent the discontinuity characteristics of approximate solution. And then the complexity of the theoretical analysis is reduced. The existence and uniqueness of the approximate solution and the stability of the scheme are proved. The optimalorder error estimates in L2(H1) and L2(L2) norms are derived. These estimates are valid under weak restrictions on the space-time mesh, namely, without the condition kn ⩾ ch2, which is necessary in traditional space-time discontinuous Galerkin methods. Numerical experiments are presented to verify the theoretical results.