Multigrid Methods for Finite Elements
Multigrid Methods for Finite Elements combines two rapidly developing fields: finite element methods, and multigrid algorithms. At the theoretical level, Shaidurov justifies the rate of convergence of various multigrid algorithms for self-adjoint and non-self-adjoint problems, positive definite and indefinite problems, and singular and spectral problems. At the practical level these statements are carried over to detailed, concrete problems, including economical constructions of triangulations and effective work with curvilinear boundaries, quasilinear equations and systems. Great attention is given to mixed formulations of finite element methods, which allow the simplification of the approximation of the biharmonic equation, the steady-state Stokes, and Navier--Stokes problems.
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Elliptic boundaryvalue problems and BubnovGalerkin method
General properties of finite elements
On the convergence of approximate solutions
5 other sections not shown
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accuracy angle apply approximate solution arbitrary vector arithmetic operations Assume assumptions of Theorem basis functions biharmonic equation bilinear form boundary value Bubnov-Galerkin method Bubnov-Galerkin system cell coefficients compute conditions G constant construct convergence corresponding curvilinear defined degrees of freedom Denote differential problem Dirichlet problem discrete problem domain eigenfunctions eigenvalues eigenvectors element of degree elliptic equal error finite element method FMG-algorithm follows Friedrichs inequality function ph Green's formula holds initial guess integer interpolation L2-norm Lagrange element Lemma linear span Lipschitz-continuity matrix Lh multigrid algorithms multigrid methods nodes notation number of arithmetic obtained orthogonal piecewise linear polynomials positive definite positive-definite prolongation Proof prove quadrature formula quasi-solution rectangle result Richardson extrapolation right-hand side satisfying the estimate scalar product section 4.6 simplex smooth solution uh solving space spectral problem subdivision subsection subspace symmetric tetrahedron three-dimensional transformation triangles two-dimensional unique solution valid vector Wh vertices W-cycle