A Time Discontinuous Galerkin Finite Element Method for Quasi-Linear Sobolev Equations

We present a time discontinuous Galerkin finite element scheme for quasi-linear Sobolev equations. The approximate solution is sought as a piecewise polynomial of degree in time variable at most q - 1 with coefficients in finite element space. This piecewise polynomial is not necessarily continuous at the nodes of the partition for the time interval. The existence and uniqueness of the approximate solution are proved by use of Brouwer's fixed point theorem. An optimal L-infinity(0, T;H-1(Omega))-norm error estimate is derived. Just because of a damping term u(xxt) included in quasi-linear Sobolev equations, which is the distinct character different from parabolic equation, more attentions are paid to this term in the study. This is the significance of this paper.