Journal ArticleDOI
A New Convergence Proof for the Multigrid Method Including the V-Cycle
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TLDR
The presented proof applies to procedures with any number of smoothing iterations and to the V-cycle and proves convergence under natural assumptions on the discretization and the elliptic problem.Abstract:
For a positive definite finite element equation we describe a multigrid iteration and prove convergence under natural assumptions on the discretization and the elliptic problem. Hitherto existing convergence proofs require a sufficiently large number of smoothing iterations and exclude the “V-cycle”. The presented proof applies to procedures with any number of smoothing iterations and to the V-cycle.read more
Citations
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Journal ArticleDOI
Multiquadrics--a scattered data approximation scheme with applications to computational fluid-dynamics-- ii solutions to parabolic, hyperbolic and elliptic partial differential equations
TL;DR: In this paper, the authors used MQ as the spatial approximation scheme for parabolic, hyperbolic and the elliptic Poisson's equation, and showed that MQ is not only exceptionally accurate, but is more efficient than finite difference schemes which require many more operations to achieve the same degree of accuracy.
Journal ArticleDOI
Finite element exterior calculus, homological techniques, and applications
TL;DR: Finite element exterior calculus as mentioned in this paper is an approach to the design and understand- ing of finite element discretizations for a wide variety of systems of partial differential equations, which brings to bear tools from differential geometry, algebraic topology, and homological algebra to develop discretiza- tions which are compatible with the geometric, topological and algebraic structures which underlie well-posedness of the PDE problem being solved.
Journal ArticleDOI
Multigrid in H(div) and H(curl)
TL;DR: If appropriate finite element spaces and appropriate additive or multiplicative Schwarz smoothers are used, then the multigrid V-cycle is an efficient solver and preconditioner for the discrete operator.
Journal ArticleDOI
The method of alternating projections and the method of subspace corrections in Hilbert space
Jinchao Xu,Ludmil T. Zikatanov +1 more
TL;DR: In this paper, the authors studied the convergence rate of alternating projections and subspace corrections in a Hilbert space setting and in particular presented a new identity for the product of nonexpansive operators that gives a sharpest possible estimate of convergence rate.
Journal ArticleDOI
Convergence estimates for multigrid algorithms without regularity assumptions
TL;DR: A new technique for proving rate of convergence estimates of multi- grid algorithms for asymmetric positive definite problems for symmetricpositive definite problems will be given in this paper.
References
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Journal ArticleDOI
An optimal order process for solving finite element equations
Randolph E. Bank,Todd F. Dupont +1 more
TL;DR: In this article, a k-level iterative procedure for solving the algebraic equations which arise from the finite element approximation of elliptic boundary value problems is presented and analyzed, and the work estimate for this procedure is proportional to the number of unknowns, an optimal order result.
Journal ArticleDOI
On the ² convergence of an algorithm for solving finite element equations
TL;DR: It is proved that the iterative method can produce a solution to the equations in O(N) arithmetical operations where N is the number of unknowns.
Book ChapterDOI
The convergence rate of a multigrid method with Gauss-Seidel relaxation for the Poisson equation
TL;DR: The numerical solution of the Poisson equation is treated by a multigrid method for a uniform grid with smoothing effect of the Gaus-Seidel relaxation by a discrete seminorm which is weaker than the energy norm.