Analytical Aspects in the Discretization of Nonlinear PDEs

The classical approach to the numerical analysis of approximation schemes for partial differential equations consists in the concept that stability and consistency imply convergence. For nonlinear equations often uniqueness and regularity properties are not available and weaker methods have to be employed to justify numerical schemes. The use of modern analytical techniques in the numerical analysis of nonlinear partial differential equations is discussed for certain classes of constrained or nondifferentiable minimization problems and nonsmooth evolution equations. This allows to prove the asymptotic accuracy of the proposed numerical methods. The application to particular instances such as the large deformation of thin elastic objects, the denoising of images, and rate-independent, inelastic material behavior provide empirical evidence of the qualitative accuracy and practical efficiency of the devised algorithms.