Interior penalty discontinuous Galerkin methods with implicit time-integration techniques for nonlinear parabolic equations (Q2846185)

The authors consider a nonlinear parabolic equation that describes the diffusion of a chemical species of concentration \(u\) in a porous medium with a source term \(f(x,u)\). For the approximation of solutions they consider interior penalty discontinuous Galerkin (IPDG) methods. The article addresses issues such as the existence of a solution, numerical stability, error estimates, and numerical performance of the methods. The first part of the article contains basic formulations and approximation properties of the IPDG methods in semidiscrete form with a \(\theta\) scheme in time. Results concerning the existence of a fully discrete solution are proven in the next part. The authors subsequently give a new stability result for fully discrete IPDG schemes. For the implicit symmetric interior penalty Galerkin method, where the time derivative is discretized by using a \(\theta\) scheme, the authors give optimal error estimates. Numerical results illustrating the performance of the methods are presented in the last section. Both backward Euler and Crank-Nicolson discretizations in time are used in the numerical examples.