COMPUTATIONAL CHALLENGES IN THE CALCULUS OF VARIATIONS

Abstract. The computational nonlinear PDEs involve minimisation problems with various striking challenges such as measure-valued solution concepts or ghost solutions. The presenta-tion starts with a class of degenerate convex minimisation problems and convergent adaptive finite element methods. The motivation for those comes from computational microstructures in case of a sufficient convexification. Finte plasticity is one example where the quasicon-vexification is not known in closed for and hence numerical relaxation is required. Polyconvex minimisation problems and their finite element analysis is discussed as well as some L1 penalty finite element method. The Mania example for the Lavrentiev phenomena for singular min-imisers with convergence towards the wrong solution studied as well as its remedy via the penalty finite element method. References [1] C. Carstensen and K. Jochimsen, Adaptive finite element methods for microstructures? Numerical experiments for a 2-well benchmark. Computing 2003. [2] C. Carstensen and Georg Dolzmann. An a priori error estimate for finite element discretisations in nonlinear elasticity for polyconvex materials under small loads. Numerische Mathematik 2004