Numerical approximation of partial differential equations

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This book focuses on the numerical approximation of partial differential equations (PDEs), aiming to provide a comprehensive illustration of numerical methods, particularly those derived from the variational formulation of PDEs. It includes stability and convergence analysis, error bounds, and algorithmic implementation discussions. The content balances theoretical analysis, algorithm descriptions, and application discussions, addressing a variety of problems—linear and nonlinear, steady and time-dependent, with both smooth and non-smooth solutions. In addition to model equations, it examines several (initial-) boundary value problems relevant to various fields. Part I focuses on the general numerical methods for discretizing PDEs, developing a thorough theory of Galerkin methods and their variants (Petrov Galerkin and generalized Galerkin), as well as collocation methods for spatial discretization. This theory is further specified for two significant numerical subspace realizations: the finite element method (including conforming, non-conforming, mixed, and hybrid types) and the spectral method (involving Legendre and Chebyshev expansions).