Numerical Approximation of Partial Differential Equations
Finite element methods for approximating partial differential equations have reached a high degree of maturity, and are an indispensible tool in science and technology. This textbook aims at providing a thorough introduction to the construction, analysis, and implementation of finite element methods for model problems arising in continuum mechanics. The first part of the book discusses elementary properties of linear partial differential equations along with their basic numerical approximation, the functional-analytical framework for rigorously establishing existence of solutions, and the construction and analysis of basic finite element methods. The second part is devoted to the optimal adaptive approximation of singularities and the fast iterative solution of linear systems of equations arising from finite element discretizations. In the third part, the mathematical framework for analyzing and discretizing saddle-point problems is formulated, corresponding finte element methods are analyzed, and particular applications including incompressible elasticity, thin elastic objects, electromagnetism, and fluid mechanics are addressed. The book includes theoretical problems and practical projects for all chapters, and an introduction to the implementation of finite element methods.
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algorithm Assume bilinear form boundary value problem bounded Cauchy–Schwarz inequality compute constant Crouzeix–Raviart deduce defined Definition Delta_x difference quotient Dirichlet boundary conditions discrete edges elementwise end end error estimate Euler scheme exact solution finite element methods finite element space fNodes fprintf fid function val Galerkin approximation grads Grads_T grid H(div heat equation implementation implies inequality inf-sup condition iteration Lemma linear system Lipschitz domain Mathematics MATLAB mesh-size nodal basis nodes norm numerical partial differential equation Poincaré inequality Poisson problem preconditioner Proof Exercise Proposition Prove refinement Remark right-hand side saddle-point problem satisfies sequence Show shown in Fig ſº Springer stability subspace symmetric system of equations theorem triangulation vector field wave equation weak formulation weak solution zeros ctr_max