Numerical Methods for Nonlinear Elliptic Differential Equations

Abstract

Nonlinear elliptic problems play an increasingly important role in mathematics, science and engineering, and create an exciting interplay. Other books discuss nonlinearity by a very few important examples. This is the first and only book, proving in a systematic and unifying way, stability and convergence results and methods for solving nonlinear discrete equations via discrete Newton methods for the different numerical methods for all these problems. The proofs use linearization, compact perturbation of the coercive principal parts, or monotone operator techniques, and approximation theory. This is examplified for linear to fully nonlinear problems (highest derivatives occur nonlinearly) and for the most important space discretization methods: conforming and nonconforming finite element, discontinuous Galerkin, finite difference and wavelet methods. The proof of stability for nonconforming methods employs the anticrime operator as an essential tool. For all these methods approximate evaluation of the discrete equations, and eigenvalue problems are discussed. The numerical methods are based upon analytic results for this wide class of problems, guaranteeing existence, uniqueness and regularity of the exact solutions. In the next book, spectral, mesh‐free methods and convergence for bifurcation and center manifolds for all these combinations are proved. Specific long open problems, solved here, are numerical methods for fully nonlinear elliptic problems, wavelet and mesh‐free methods for nonlinear problems, and more general nonlinear boundary conditions. Adaptivity is discussed for finite element and wavelet methods with totally different techniques.