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An Odyssey Into Local Refinement And Multilevel Preconditioning I: Optimality Of . . .
 SIAM J. NUMER. ANAL
, 2002
"... In this article, we examine the BramblePasciakXu (BPX) preconditioner in the setting of local 2D and 3D mesh refinement. While the available optimality results for the BPX preconditioner have been constructed primarily in the setting of uniformly refined meshes, a notable exception is the 2D resul ..."
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Cited by 26 (14 self)
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In this article, we examine the BramblePasciakXu (BPX) preconditioner in the setting of local 2D and 3D mesh refinement. While the available optimality results for the BPX preconditioner have been constructed primarily in the setting of uniformly refined meshes, a notable exception is the 2D result due to Dahmen and Kunoth, which established BPX optimality on meshes produced by a restricted class of local 2D redgreen refinement. The purpose of this article is to extend the original 2D DahmenKunoth result to several additional types of local 2D and 3D redgreen (conforming) and red (nonconforming) refinement procedures. The extensions are accomplished through a 3D extension of the 2D framework in the original DahmenKunoth work, by which the question of optimality is reduced to establishing that locally enriched finite element subspaces allow for the construction of a scaled basis which is formally Riesz stable. This construction in turn rests entirely on establishing a number of geometrical properties between neighboring simplices produced by the local refinement algorithms. These properties are then used to build Rieszstable scaled bases for use in the BPX optimality framework. Since the theoretical framework supports arbitrary spatial dimension d 1, we indicate clearly which geometrical properties, established here for several 2D and 3D local refinement procedures, must be reestablished to show BPX optimality for spatial dimension 4. Finally, we also present a simple alternative optimality proof of the BPX preconditioner on quasiuniform meshes in two and three spatial dimensions, through the use of Kfunctionals and H stability of L 2 projection for s 1. The proof techniques we use are quite general; in particular, the results require no smoothnes...
On additive Schwarz preconditioners for sparse grid discretizations
 NUMER. MATH
, 1994
"... Based on the framework of subspace splitting and the additive Schwarz scheme, we give bounds for the condition number of multilevel preconditioners for sparse grid discretizations of elliptic model problems. For a BXPlike preconditioner we derive an estimate of the optimal order O(1) and for a HB ..."
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Cited by 25 (9 self)
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Based on the framework of subspace splitting and the additive Schwarz scheme, we give bounds for the condition number of multilevel preconditioners for sparse grid discretizations of elliptic model problems. For a BXPlike preconditioner we derive an estimate of the optimal order O(1) and for a HBlike variant we obtain an estimate of the order O(k2 · 2k/2), where k denotes the number of levels employed. Furthermore, we confirm these results by numerically computed condition numbers.
On Two Ways Of Stabilizing The Hierarchical Basis Multilevel Methods
 SIAM Review
, 1997
"... A survey of two approaches for stabilizing the hierarchical basis (HB) multilevel preconditioners, both additive and multiplicative, is presented. The first approach is based on the algebraic extension of the twolevel methods, exploiting recursive calls to coarser discretization levels. These recur ..."
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Cited by 23 (5 self)
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A survey of two approaches for stabilizing the hierarchical basis (HB) multilevel preconditioners, both additive and multiplicative, is presented. The first approach is based on the algebraic extension of the twolevel methods, exploiting recursive calls to coarser discretization levels. These recursive calls can be viewed as inner iterations (at a given discretization level), exploiting the already defined preconditioner at coarser levels in a polynomiallybased inner iteration method. The latter gives rise to hybridtype multilevel cycles. This is the socalled (hybrid) algebraic multilevel iteration (AMLI) method. The second approach is based on a different direct multilevel splitting of the finite element discretization space. This gives rise to the socalled wavelet multilevel decomposition based on L 2 projections, which in practice must be approximated. Both approachesthe AMLI one and the one based on approximate wavelet decompositionslead to optimal relative condition numbers of the multilevel preconditioners.
Adaptive Multilevel Techniques for Mixed Finite Element Discretizations of Elliptic Boundary Value Problems
 SIAM J. NUMER. ANAL
, 1994
"... We consider mixed finite element discretizations of linear second order elliptic boundary value problems with respect to an adaptively generated hierarchy of possibly highly nonuniform simplicial triangulations. By a well known postprocessing technique the discrete problem is equivalent to a modifie ..."
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Cited by 21 (7 self)
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We consider mixed finite element discretizations of linear second order elliptic boundary value problems with respect to an adaptively generated hierarchy of possibly highly nonuniform simplicial triangulations. By a well known postprocessing technique the discrete problem is equivalent to a modified nonconforming discretization which is solved by preconditioned cgiterations using a multilevel BPXtype preconditioner designed for standard nonconforming approximations. Local refinement of the triangulations is based on an a posteriori error estimator which can be easily derived from superconvergence results. The performance of the preconditioner and the error estimator is illustrated by several numerical examples.
FieldOfValues Analysis Of Preconditioned Iterative Methods For Nonsymmetric Elliptic Problems
 Numer. Math
, 1997
"... . The convergence rate of Krylov subspace methods for the solution of nonsymmetric systems of linear equations, such as GMRES or FOM, is studied. Bounds on the convergence rate are presented which are based on the smallest real part of the field of values of the coefficient matrix and of its inverse ..."
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Cited by 21 (0 self)
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. The convergence rate of Krylov subspace methods for the solution of nonsymmetric systems of linear equations, such as GMRES or FOM, is studied. Bounds on the convergence rate are presented which are based on the smallest real part of the field of values of the coefficient matrix and of its inverse. Estimates for these quantities are available during the iteration from the underlying Arnoldi process. It is shown how these bounds can be used to study the convergence properties, in particular, the dependence on the meshsize and on the size of the skewsymmetric part, for preconditioners for finite element discretizations of nonsymmetric elliptic boundary value problems. This is illustrated for the hierarchical basis and multilevel preconditioners which constitute popular preconditioning strategies for such problems. Key words. Krylov subspace methods, GMRES, FOM, field of values, hierarchical basis, multilevel preconditioning, nonsymmetric elliptic problems 1. Introduction. The subjec...
Optimality of multilevel preconditioners for local mesh refinement in three dimensions
 SIAM J. Numer. Anal
"... Abstract. In this article, we establish optimality of the Bramble–Pasciak–Xu (BPX) norm equivalence and optimality of the wavelet modified (or stabilized) hierarchical basis (WHB) preconditioner in the setting of local 3D mesh refinement. In the analysis of WHB methods, a critical first step is to e ..."
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Cited by 18 (9 self)
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Abstract. In this article, we establish optimality of the Bramble–Pasciak–Xu (BPX) norm equivalence and optimality of the wavelet modified (or stabilized) hierarchical basis (WHB) preconditioner in the setting of local 3D mesh refinement. In the analysis of WHB methods, a critical first step is to establish the optimality of BPX norm equivalence for the refinement procedures under consideration. While the available optimality results for the BPX norm have been constructed primarily in the setting of uniformly refined meshes, a notable exception is the local 2D redgreen result due to Dahmen and Kunoth. The purpose of this article is to extend this original 2D optimality result to the local 3D redgreen refinement procedure introduced by Bornemann, Erdmann, and Kornhuber, and then to use this result to extend the WHB optimality results from the quasiuniform setting to local 2D and 3D redgreen refinement scenarios. The BPX extension is reduced to establishing that locally enriched finite element subspaces allow for the construction of a scaled basis which is formally Riesz stable. This construction turns out to rest not only on the shape regularity of the refined elements, but also critically on a number of geometrical properties we establish between neighboring simplices produced by the Bornemann–Erdmann–Kornhuber (BEK) refinement procedure. It is possible to show that the number of degrees of freedom used for smoothing is bounded by a constant times the number of degrees of freedom introduced at that level of refinement, indicating that a practical, implementable version of the resulting BPX preconditioner for the BEK refinement setting has provably optimal (linear) computational complexity per iteration. An interesting implication of the optimality of the WHB preconditioner is the a priori H 1stability of the L2projection. The existing a posteriori approaches in the literature dictate a reconstruction of the mesh if such conditions cannot be satisfied. The theoretical framework employed supports arbitrary spatial dimension d ≥ 1 and requires no coefficient smoothness assumptions beyond those required for wellposedness in H 1.
A Robust Hierarchical Basis Preconditioner On General Meshes
 Numer. Math
, 1995
"... . In this paper, we introduce a multilevel direct sum space decomposition of general, possibly locally refined linear or multilinear finite element spaces. In contrast to the wellknown BPX and hierarchical basis preconditioners, the corresponding additive Schwarz preconditioner will be robust for ..."
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Cited by 17 (0 self)
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. In this paper, we introduce a multilevel direct sum space decomposition of general, possibly locally refined linear or multilinear finite element spaces. In contrast to the wellknown BPX and hierarchical basis preconditioners, the corresponding additive Schwarz preconditioner will be robust for a class of singularly perturbed elliptic boundary value problems. Important for an efficient implementation is that stable bases of the subspaces defining our decomposition, consisting of functions having small supports can be easily constructed. 1. Background and motivation This paper deals with additive Schwarz multilevel preconditioners for solving symmetric second order linear elliptic boundary value problems (cf. [Xu92, Yse93, GO95a]). We assume a nested sequence of linear or multilinear finite element spaces M 0 ae M 1 ae : : : ae M J ae : : : and discretize the boundary value problem on M J using Galerkin's method. Additive Schwarz preconditioners are based on a subspace decomposi...
Rate of Convergence for some constraint decomposition methods for nonlinear variational inequalities
 NUMER. MATH. (2003) 93: 755–786
, 2003
"... ..."
Multigrid methods for obstacle problems
"... Abstract. In this review, we intend to clarify the underlying ideas and the relations between various multigrid methods ranging from subset decomposition, to projected subspace decomposition and truncated multigrid. In addition, we present a novel globally convergent inexact active set method which ..."
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Cited by 14 (2 self)
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Abstract. In this review, we intend to clarify the underlying ideas and the relations between various multigrid methods ranging from subset decomposition, to projected subspace decomposition and truncated multigrid. In addition, we present a novel globally convergent inexact active set method which is closely related to truncated multigrid. The numerical properties of algorithms are carefully assessed by means of a degenerate problem and a problem with a complicated coincidence set. 1.