A Multi-Level Iterative Method for Solving Finite Element Equations

Bank, Randolph E.; Sherman, Andrew J.

doi:10.2523/7683-ms

Cited by 16 publications

(28 citation statements)

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“…Some works pay little attention to that point. The key argument is that A 11 has condition number O(1) [1,5,10], so that any reasonable iterative scheme converges quickly in a number of steps independent of the grid size. In practice however, the cost of such inner iterations becomes rapidly prohibitive.…”

Section: Introductionmentioning

confidence: 99%

Using approximate inverses in algebraic multilevel methods

Notay

1998

Numerische Mathematik

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This paper deals with the iterative solution of large sparse symmetric positive definite systems. We investigate preconditioning techniques of the two-level type that are based on a block factorization of the system matrix. Whereas the basic scheme assumes an exact inversion of the submatrix related to the first block of unknowns, we analyze the effect of using an approximate inverse instead. We derive condition number estimates that are valid for any type of approximation of the Schur complement and that do not assume the use of the hierarchical basis. They show that the two-level methods are stable when using approximate inverses based on modified ILU techniques, or explicit inverses that meet some row-sum criterion. On the other hand, we bring to the light that the use of standard approximate inverses based on convergent splittings can have a dramatic effect on the convergence rate. These conclusions are numerically illustrated on some examples.

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Section: Introductionmentioning

confidence: 99%

Using approximate inverses in algebraic multilevel methods

Notay

1998

Numerische Mathematik

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“…The next lemma shows that a bound on the interpolant as in (16) in essentially equivalent to a strengthened Cauchy inequality [3] [12]. Proof.…”

Section: Theorem 43 Let U ∈ M and Let > K Thenmentioning

confidence: 99%

“…Using this paradigm, in Sect. 3 we develop a practical algorithm for creating a logical refinement structure given only the final fine mesh. This is done by a process of coarsening, in which one vertex is removed at each step, and the mesh updated.…”

Section: Introductionmentioning

confidence: 99%

An algorithm for coarsening unstructured meshes

Bank

1996

Numerische Mathematik

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We develop and analyze a procedure for creating a hierarchical basis of continuous piecewise linear polynomials on an arbitrary, unstructured, nonuniform triangular mesh. Using these hierarchical basis functions, we are able to define and analyze corresponding iterative methods for solving the linear systems arising from finite element discretizations of elliptic partial differential equations. We show that such iterative methods perform as well as those developed for the usual case of structured, locally refined meshes. In particular, we show that the generalized condition numbers for such iterative methods are of order J 2 , where J is the number of hierarchical basis levels.

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“…(1) can be regarded as being independent of 2) and then for simultaneous solution for u and ).. The many details omitted here can be found in [Bank 1985[Bank , 1986Bank and Chan 1986;Bank and Douglas 1985;Bank and Dupont 1981;Bank and Rose 1982;Bank and Weiser 1985;Bank et al 1983;Chan and Keller 1982] and the references contained therein. The process terminates when the sum of the squares of the estimated solution errors on all triangular elements in a grid is sufficiently small or a maximum number of grids has been created.…”

Section: Pltmg Partial Differential Equation Packagementioning

confidence: 99%

“…It incorporates solution error estimation and automatic local refinement of unstructured meshes in a piecewise linear finite element method [Bank and Weiser 1985;Bank 1986;Bank et al 1983]; approximate Newton and continuation methods [Bank and Chan *The author gratefully acknowledges support through NSF Grant No. ASC-8519354. 1986; Bank and Rose 1982;Chan and Keller 1982] for resulting nonlinear algebraic equations; and multigrid linear algebra techniques [Bank and Douglas 1985;Bank and Dupont 1981] incorporating a version of the Yale Sparse Matrix Package [Eisenstat et al 1982]. Use of such a code as a testbed to study multiprocessing performance differs from approaches focussing on application-specific software and from those directing attention to single important components of such software (e.g., [Chen et al 1984;Seager 1986]), both in its objectives and what it can achieve.…”

Section: Introductionmentioning

confidence: 99%

Microtasking general purpose partial differential equation software on the CRAY X-MP

Bieterman

1988

J Supercomput

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Microtasking on four-processor CRAY X-MP computers is examined. The examination includes descriptions of important properties of Cray microtasking and an explanation of how to apply it to a program package. The experience of applying Cray microtasking to the general purpose partial differential equation package PLTMG of R. Bank et al. [1985, 1986] is communicated. Results of experiments in a single-job (dedicated) computing environment show that a large part ofPLTMG's interprocedural parallelism can be exploited with microtasking and little to moderate human effort. An experiment involving PLTMG and conducted to assess the performance of Cray microtasking in a multiprogramming (batch) environment is also described. This and related experiments described in [Bieterman 1987] are evidently the first of their kind reported.

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A Multi-Level Iterative Method for Solving Finite Element Equations

Cited by 16 publications

References 0 publications

Using approximate inverses in algebraic multilevel methods

Using approximate inverses in algebraic multilevel methods

An algorithm for coarsening unstructured meshes

Microtasking general purpose partial differential equation software on the CRAY X-MP

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