Discontinuous Galerkin time-stepping method for semilinear parabolic problems with mild growth condition

We investigate the numerical solution of semilinear parabolic equations under mild growth conditions on the nonlinear source function, focusing on time discretization using the discontinuous Galerkin (DG) method. A novel technique is proposed for handling terms that arise from mild growth condition during the application of the DG method in time. This approach ensures the solvability of the semi-discrete scheme without requiring any boundedness assumptions on the nonlinearity. Additionally, we establish the optimal order of convergence for the semi-discrete approach. Subsequently, we combine the continuous Galerkin method in space with the discontinuous Galerkin method in the temporal direction to develop a fully discrete scheme. We demonstrate that the fully discrete (DG-CG) scheme achieves optimal convergence rates while accommodating the mild growth conditions under various hypotheses on the exact solution u. Our theoretical findings are verified through a series of numerical experiments.