A new approach for solving stokes systems arising from a distributive relaxation method

The distributed relaxation method for the Stokes problem has been advertised as an adequate change of variables that leads to a lower triangular system with Laplace operators on the main diagonal for which multigrid methods are very efficient. We show that under high regularity of the Laplacian, the transformed system admits almost block‐lower triangular form. We analyze the distributed relaxation method and compare it with other iterative methods for solving the Stokes system. We also present numerical experiments illustrating the effectiveness of the transformation which is well established for certain finite difference discretizations of Stokes problems. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 898–914, 2011     

Improving solve time of aggregation‐based adaptive AMG

This paper proposes improving the solve time of a bootstrap algebraic multigrid (AMG) designed previously by the authors. This is achieved by incorporating the information, a set of algebraically smooth vectors, generated by the bootstrap algorithm, in a single hierarchy by using sufficiently large aggregates, and these aggregates are compositions of aggregates already built throughout the bootstrap algorithm. The modified AMG method has good convergence properties and shows significant reduction in both memory and solve time. These savings with respect to the original bootstrap AMG are illustrated on some difficult (for standard AMG) linear systems arising from discretization of scalar and vector function elliptic partial differential equations in both 2D and 3D.     

Incomplete block-matrix factorization iterative methods for convection-diffusion problems

Standard Galerkin finite element methods or finite difference methods for singular perturbation problems lead to strongly unsymmetric matrices, which furthermore are in general notM-matrices. Accordingly, preconditioned iterative methods such as preconditioned (generalized) conjugate gradient methods, which have turned out to be very successful for symmetric and positive definite problems, can fail to converge or require an excessive number of iterations for singular perturbation problems.This is not so much due to the asymmetry, as it is to the fact that the spectrum can have both eigenvalues with positive and negative real parts, or eigenvalues with arbitrary small positive real parts and nonnegligible imaginary parts. This will be the case for a standard Galerkin method, unless the meshparameterh is chosen excessively small. There exist other discretization methods, however, for which the corresponding bilinear form is coercive, whence its finite element matrix has only eigenvalues with positive real parts; in fact, the real parts are positive uniformly in the singular perturbation parameter.In the present paper we examine the streamline diffusion finite element method in this respect. It is found that incomplete block-matrix factorization methods, both on classical form and on an inverse-free (vectorizable) form, coupled with a general least squares conjugate gradient method, can work exceptionally well on this type of problem. The number of iterations is sometimes significantly smaller than for the corresponding almost symmetric problem where the velocity field is close to zero or the singular perturbation parameter =1.The 2 ^nd author's research was sponsored by Control Data Corporation through its PACER fellowship program.The 3 ^rd author's research was supported by the Netherlands organization for scientific research (NWO).On leave from the Institute of Mathematics, Academy of Science, 1090 Sofia, P.O. Box 373, Bulgaria.     

An element agglomeration nonlinear additive Schwarz preconditioned Newton method for unstructured finite element problems

This paper extends previous results on nonlinear Schwarz preconditioning (Cai and Keyes 2002) to unstructured finite element elliptic problems exploiting now nonlocal (but small) subspaces. The nonlocal finite element subspaces are associated with subdomains obtained from a non-overlapping element partitioning of the original set of elements and are coarse outside the prescribed element subdomain. The coarsening is based on a modification of the agglomeration based AMGe method proposed in Jones and Vassilevski 2001. Then, the algebraic construction from Jones, Vassilevski and Woodward 2003 of the corresponding non-linear finite element subproblems is applied to generate the subspace based nonlinear preconditioner. The overall nonlinearly preconditioned problem is solved by an inexact Newton method. A numerical illustration is also provided.This work was performed under the auspices of the U.S. Department of Energy by the University of California Lawrence Livermore National Laboratory: contract/grant number: W-7405-Eng-48. The contribution of the second author was also partially supported by Polish Scientific Grant 2/P03A/005/24.     

Arsenic in marine tissues — The challenging problems to electrothermal and hydride generation atomic absorption spectrometry

Analytical problems in determination of arsenic in marine tissues are addressed. Procedures for the determination of total As in solubilized or extracted tissues with tetramethylammonium hydroxide and methanol have been elaborated. Several typical lyophilized tissues were used: NIST SRM 1566a ‘Oyster Tissue’, BCR-60 CRM ‘Trace Elements in an Aquatic Plant (Lagarosiphon major)’, BCR-627 ‘Forms of As in Tuna Fish Tissue’, IAEA-140/TM ‘Sea Plant Homogenate’, NRCC DOLT-1 ‘Dogfish Liver’ and two representatives of the Black Sea biota, Mediterranean mussel (Mytilus galloprovincialis) and Brown algae (Cystoseira barbata). Tissues (nominal 0.3 g) were extracted in tetramethylammonium hydroxide (TMAH) 1 ml of 25% m/v TMAH and 2 ml of water) or 5 ml of aqueous 80% v/v methanol (MeOH) in closed vessels in a microwave oven at 50 °C for 30 min. Arsenic in solubilized or extracted tissues was determined by electrothermal atomic absorption spectrometry (ETAAS) after appropriate dilution (nominally to 25 ml, with further dilution as required) under optimal instrumental parameters (pyrolysis temperature 900 °C and atomization temperature 2100 °C) with 1.5 μg Pd as modifier on Zr–Ir treated platform. Platforms have been pre-treated with 2.7 μmol of zirconium and then with 0.10 μmol of iridium which served as a permanent chemical modifier in direct ETAAS measurements and as an efficient hydride sequestration medium in flow injection hydride generation (FI-HG)–ETAAS. TMAH and methanol extract 96–108% and 51–100% of As from CRMs. Various calibration approaches have been considered and critically evaluated. The effect of species-dependent slope of calibration graph or standard additions plot for total As determination in a sample comprising of several individual As species with different ETAAS behavior has been considered as a kind of ‘intrinsic element speciation interference’ that cannot be completely overcome by standard additions technique. Calibration by means of CRMs has given only semi-quantitative results. The limits of detection (3σ) were in the range 0.5–1.2 mg kg^− 1 As dry weight (wt.) for direct ETAAS analysis of extracts in both TMAH and MeOH. Within-run precision (RSD%) was 5–15% and 7–20% for TMAH and MeOH extracts at As levels 4–50 mg kg^− 1 dry wt., respectively.     

Least-squares mixed finite element methods for non-selfadjoint elliptic problems: I. Error estimates

Summary. A least-squares mixed finite element method for general second-order non-selfadjoint elliptic problems in two- and three-dimensional domains is formulated and analyzed. The finite element spaces for the primary solution approximation and the flux approximation consist of piecewise polynomials of degree and respectively. The method is mildly nonconforming on the boundary. The cases and are studied. It is proved that the method is not subject to the LBB-condition. Optimal - and -error estimates are derived for regular finite element partitions. Numerical experiments, confirming the theoretical rates of convergence, are presented. Received October 15, 1993 / Revised version received August 2, 1994     

Interior penalty preconditioners for mixed finite element approximations of elliptic problems

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.

     

A survey of multilevel preconditioned iterative methods

We survey multilevel iterative methods applied for solving large sparse systems with matrices, which depend on a level parameter, such as arise by the discretization of boundary value problems for partial differential equations when successive refinements of an initial discretization mesh is used to construct a sequence of nested difference or finite element meshes.We discuss various two-level (two-grid) preconditioning techniques, including some for nonsymmetric problems. The generalization of these techniques to the multilevel case is a nontrivial task. We emphasize several ways this can be done including classical multigrid methods and a recently proposed algebraic multilevel preconditioning method. Conditions for which the methods have an optimal order of computational complexity are presented.On leave from the Institute of Mathematics, and Center for Informatics and Computer Technology, Bulgarian Academy of Sciences, Sofia, Bulgaria. The research of the second author reported here was partly supported by the Stichting Mathematisch Centrum, Amsterdam.     

Stabilizing the Hierarchical Basis by Approximate Wavelets,I: Theory

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. © 1997 by John Wiley & Sons, Ltd.     

Preconditioning Mixed Finite Element Saddle-point Elliptic Problems

We consider saddle-point problems that typically arise from the mixed finite element discretization of second-order elliptic problems. By proper equivalent algebraic operations the considered saddle-point problem is transformed to another saddle-point problem. The resulting problem can then be efficiently preconditioned by a block-diagonal matrix or by a factored block-matrix (the blocks correspond to the velocity and pressure, respectively). Both preconditioners have a block on the main diagonal that corresponds to the bilinear form (δ is a positive parameter) and a second block that is equal to a constant times the identity operator. We derive uniform bounds for the negative and positive eigenvalues of the preconditioned operator. Then any known preconditioner for the above bilinear form can be applied. We also show some numerical experiments that illustrate the convergence properties of the proposed technique.