The main goal in device and process modeling and simulation is to obtain accurate results which physically fulfill the simulation requirements. For this reason it is of great importance to find out if a simulation result is accurate or not. In the case of an inaccuracy the simulation domain has to be better resolved in the region where the errors are large. Therefore it is necessary to obtain some measurement of the local error of the simulation.
The results of a simulation are highly mesh dependent. Therefore each refinement step affects the simulation result. This effect becomes problematic in stepping simulations where, e.g., output characteristics are calculated. Each mesh refinement step causes jitter in the output characteristics.
In semiconductor simulation the finite element, the finite volume and the finite difference method are used to obtain numerical results. In the mathematical literature there are many different approaches to error estimation. The estimator which is easiest to implement is the ZZ estimator, which only takes the solution of the equation and calculates the local smoothness. For some linear elliptic and parabolic partial differential equations this method can be proven to be convergent to the correct solution.
Apart from these simple error estimators we use residual-based estimators. These estimators are only valid for one special kind of differential equation but are more reliable to converge to the analytical solution, also in nonlinear equations, when the mesh is being refined.
Another important issue in mesh adaptation is coarsement or un-refinement. For transient calculations some regions are relevant at certain times but totally irrelevant at others. The refinement leads to a pollution of the mesh and there are several irrelevant equations in the resulting equation system. For this reason the mesh generator has to provide coarsement and an estimator has to trigger the re-coarsement mechanism.
In our work we implement a posteriori error estimators for several PDEs and discretization schemes as well as refinement strategies for meshes.