This presentation concerns five topics in computational partial differential equations with the overall goals of reliable error control and efficient simulation.
The presentation is also an advertisement for nonstandard discretisations in linear and nonlinear Computational PDEs with surprising advantages over conforming
finite element schemes and the combination
of the two. The equivalence of various first-order methods is explained for the linear Poisson model problem with conforming
(CFEM), nonconforming (NC-FEM), and mixed finite element methods (MFEM) and others discontinuous Galerkin finite element (dGFEM). The Stokes
equations illustrate the NCFEM and the pseudo-stress MFEM and optimal convergence of adaptive mesh-refining as well as for guaranteed error bounds.
An optimal adaptive CFEM computation of elliptic eigenvalue
problems and the computation of guaranteed upper and lower eigenvalue bounds based on NCFEM. The obstacle problem and its guaranteed error
control follows another look due to D. Braess with guaranteed error bounds and their effectivity indices between 1 and 3. Some remarks on computational
microstructures with degenerate convex minimisation
problems conclude the presentation.