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Spectral methods have long been popular in direct and large eddy simulation of turbulent flows, but their use in areas with complex-geometry computational domains has historically been much more limited. More recently, the need to find accurate solutions to the viscous flow equations around complex configurations has led to the development of high-order discretization procedures on unstructured meshes, which are also recognized as more efficient for solution of time-dependent oscillatory solutions over long time periods. This book, an updated edition on the original text, presents the recent and significant progress in multi-domain spectral methods at both the fundamental and application level. Containing material on discontinuous Galerkin methods, non-tensorial nodal spectral element methods in simplex domains, and stabilization and filtering techniques, this text introduces the use of spectral/hp element methods with particular emphasis on their application to unstructured meshes. It provides a detailed explanation of the key concepts underlying the methods along with practical examples of their derivation and application.

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.