Numerical engineering: design of PDE black-box solvers

The design of PDE black-box solvers (for nonlinear systems of elliptic and parabolic PDEs) needs many compromises between efficiency and robustness which we call ‘Numerical Engineering’. The requirements for a black-box solver are formulated and the way how to meet them is presented, guided by many years of practical experience in the design of the program packages fidisol/cadsol, vecfem and linsol. The basic approach to the new finite difference element method (fdem) program package, an FDM on an unstructured FEM grid, is discussed. The common feature of all these methods is the error equation that allows a transparent balancing of all errors. The discretization errors are estimated from difference formulae of different consistency orders. The error balancing must include the iterative solution of the large and sparse linear systems by the linsol program package. The real challenge is the parallelization on distributed memory parallel computers which is solved by corresponding data structures with optimal communication patterns and redistribution after each grid refinement cycle.