We consider a singularly perturbed time-dependent problem with a shift term in space. On appropriately defined layer adapted meshes of Durán- and S-type we derive a-priori error estimates for the stationary problem. Using a discontinuous Galerkin method in time we obtain error estimates for the full discretisation. Introduction of a weighted scalar products and norms allows us to estimate the error of the time-dependent problem in energy and balanced norm. So far it was open to prove such a result. Error estimates in the energy norm is for the standard finite element discretization in space, and for the error estimate in the balanced norm the computation of the numerical solution is changed by using a different bilinear form. Some numerical results are given to confirm the predicted theory and to show the effect of shifts on the solution.
|Number of pages||12|
|Journal||Applied mathematics and computation|
|Publication status||Published - 15 Jan 2023|
DFG Classification of Subject Areas according to Review Boards
- Discontinuous Galerkin, Singularly perturbed, Spatial shift, Time-dependent, NUMERICAL TREATMENT, APPROXIMATION, SINGULAR PERTURBATION ANALYSIS, BOUNDARY-VALUE-PROBLEMS, FINITE-ELEMENT METHODS, CONVERGENCE, PARABOLIC PROBLEMS, DIFFERENTIAL-DIFFERENCE EQUATIONS