The Direct Solution of the Biharmonic Equation on Rectangular Regions and the Poisson Equation on Irregular Regions

Dedicated to ProfessorJoachim Nitsche on the occasion of the sixtieth anniversary of his birthday. Abstract.We consider the problem of solving the algebraic system of equations which arise from the discretization of symmetric elliptic boundary value problems via finite element methods. A new class of preconditioners for these discrete systems is developed based on substructuring (also known as domain decomposition). The resulting preconditioned algorithms are well suited to emerging parallel computing architectures.The proposed methods are applicable to problems on general domains involving differential operators with rather general coefficients. A basic theory for the analysis of the condition number of the preconditioned system (which determines the iterative convergence rate of the algorithm) is given. Techniques for applying the theory and algorithms to problems with irregular geometry are discussed and the results of extensive numerical experiments are reported.1. Introduction. The aim of this series of papers is to propose and analyze methods for efficiently solving the equations resulting from finite element discretizations of second-order elliptic boundary value problems on general domains in R2 and R3. In particular, we shall be concerned with constructing easily invertible and "effective" preconditioners for the resulting system of discrete equations which can be used in a preconditioned iterative algorithm to achieve a rapid solution method. The methods to be presented are well suited to parallel computing architectures.In this paper we shall restrict ourselves to boundary value problems in R2. Let fî be a bounded domain in R2 with a piecewise smooth boundary dû. As a model problem for a second-order uniformly elliptic equation we shall consider the

show abstract

“…The system (1.7) is symmetric with respect to the inner product defined by TV (1)(2)(3)(4)(5)(6)(7)(8) [a,ß]= ¿ZBijOißj.…”

Section: Introductionmentioning

confidence: 99%

The construction of preconditioners for elliptic problems by substructuring. I

Bramble¹,

Pasciak²,

Schatz³

1986

Math. Comp.

397

173

View full text Add to dashboard Cite

show abstract

“…We emphasize that the purpose here is to construct a preconditioner for the original problem which in our case is more general. 4. Applications and Numerical Experiments.…”

Section: ä2(vhvh) < 2ä2(vh -Vh Vh -Vh) + 2ä2(vh V")mentioning

confidence: 99%

“…For other works dealing with the solution of boundary value problems via substructuring cf. [4], [5],…”

mentioning

confidence: 99%

See 1 more Smart Citation

An Iterative Method for Elliptic Problems on Regions Partitioned into Substructures

Bramble¹,

Pasciak²,

Schatz³

1986

Mathematics of Computation

View full text Add to dashboard Cite

Abstract. Some new preconditioners for discretizations of elliptic boundary problems are studied. With these preconditioners, the domain under consideration is broken into subdomains and preconditioners are defined which only require the solution of matrix problems on the subdomains. Analytic estimates are given which guarantee that under appropriate hypotheses, the preconditioned iterative procedure converges to the solution of the discrete equations with a rate per iteration that is independent of the number of unknowns. Numerical examples are presented which illustrate the theoretically predicted iterative convergence rates.

show abstract

“…The usefulness of these algorithms has been extended in recent years to problems on general bounded regions by the development of capacitance matrix, or imbedding, methods; see Buzbee and Dorr [6], Buzbee, Dorr, George and Golub [7] [42], [43], Proskurowski and Widlund [44], [45], Shieh [46], [47], [48] and Widlund [57]. We refer to Proskurowski and Widlund [44] for a discussion of this development up to the beginning of 1976.…”

mentioning

confidence: 99%

Untitled

1979

Math. Comp.

View full text Add to dashboard Cite

The Direct Solution of the Biharmonic Equation on Rectangular Regions and the Poisson Equation on Irregular Regions

Cited by 89 publications

References 24 publications

The construction of preconditioners for elliptic problems by substructuring. I

The construction of preconditioners for elliptic problems by substructuring. I

An Iterative Method for Elliptic Problems on Regions Partitioned into Substructures

Untitled

Contact Info

Product

Resources

About