Panayot S. Vassilevski

Finite volume methods for convection-diffusion problems

Derivation, stability, and error analysis in both discrete

Multilevel iterative methods for mixed finite element discretizations of elliptic problems

H^1

A general mixed covolume framework for constructing conservative schemes for elliptic problems

- and

Interior penalty preconditioners for mixed finite element approximations of elliptic problems

L^2

Stabilizing the Hierarchical Basis by Approximate Wavelets, I: Theory

-norms for cell-centered finite volume approximations of convection-diffusion problems are presented. Various upwind strategies are investigated. The theoretical results are illustrated by numerical examples.

Circulant block-factorization preconditioners for elliptic problems

SummaryFor solving second order elliptic problems discretized on a sequence of nested mixed finite element spaces nearly optimal iterative methods are proposed. The methods are within the general framework of the product (multiplicative) scheme for operators in a Hilbert space, proposed recently by Bramble, Pasciak, Wang, and Xu [5,6,26,27] and make use of certain multilevel decomposition of the corresponding spaces for the flux variable.

A framework for block ILU factorizations using block-size reduction

We present a general framework for the finite volume or covolume schemes developed for second order elliptic problems in mixed form, i.e., written as first order systems. We connect these schemes to standard mixed finite element methods via a one-to-one transfer operator between trial and test spaces. In the nonsymmetric case (convection-diffusion equation) we show one-half order convergence rate for the flux variable which is approximated either by the lowest order Raviart-Thomas space or by its image in the space of discontinuous piecewise constants. In the symmetric case (diffusion equation) a first order convergence rate is obtained for both the state variable (e.g., concentration) and its flux. Numerical experiments are included.

Stabilizing the Hierarchical Basis by Approximate Wavelets II: Implementation and Numerical Results

It is established that an interior penalty method applied to second-order elliptic problems gives rise to a local operator which is spectrally equivalent to the corresponding nonlocal operator arising from the mixed finite element method. This relation can be utilized in order to construct preconditioners for the discrete mixed system. As an example, a family of additive Schwarz preconditioners for these systems is constructed. Numerical examples which confirm the theoretical results are also presented.

Finite difference schemes on grids with local refinement in time and space for parabolic problems. I. Derivation, stability, and error analysis

This paper proposes a stabilization of the classical hierarchical basis (HB) method by modifying the HB functions using some computationally feasible approximate L2-projections onto finite element spaces of relatively coarse levels. The corresponding multilevel additive and multiplicative algorithms give spectrally equivalent preconditioners, and one action of such a preconditioner is of optimal order computationally. The results are regularity-free for the continuous problem (second order elliptic) and can be applied to problems with rough coefficients and local refinement.

Local refinement techniques for elliptic problems on cell-centered grids; II. Optimal order two-grid iterative methods

New circulant block-factorization preconditioners are introduced and studied. The general approach is first formulated for the case of block tridiagonal sparse matrices. Then estimates of the relative condition number for a model Dirichlet boundary value problem are derived. In the case ofy-periodic problems the circulant block-factorization preconditioner is shown to give an optimal convergence rate. Finally, using a proper imbedding of the original Dirichlet boundary value problem to ay-periodic one a preconditioner of optimal convergence rate for the general case is obtained. The total computational cost of the preconditioner isO (N logN) (based on FFT), whereN is the number of unknowns. That is, the algorithm is nearly optimal. Various numerical tests that demonstrate the features of the circulant block-factorization preconditioners are presented.ZusammenfassungNeue zyklische Matrixzerlegungen werden eingeführt und untersucht. Der allgemeine Ansatz wird für den Fall blocktridiagonaler schwachbesetzter Matrizen formuliert. Danach werden Abschätzungen der relativen Konditionszahl für ein Dirichlet-Modellproblem abgeleitet. Es wird gezeigt, daß die zyklische Matrixzerlegung im Falley-periodischer Aufgaben optimale Konvergenzraten liefert. Nach Einbettung des ursprünglichen Dirichlet-Problems in einey-periodische Aufgabe erhält man den allgemeinen Fall. Der Gesamtaufwand des Präkonditionierers beträgtO (N logN) gemäß des FFT-Aufwandes, wobeiN die Zahl der Unbekannten ist. Damit ist der Algorithmus fast optimal. Verschiedene numerische Tests zeigen die Eigenschaften der zyklischen Matrixzerlegung.