An energy-preserving explicit extension operator is proposed to extend finite element functions defined on the boundary of a star-shaped polygonal domain into its interior. The pre-assigned finite element triangulation in the interior of the domain needs not be multilevel-structured. The extension operator has wide applications in the construction of non-overlapping domain decomposition methods...

The author’s algebraic theory of boundary value problems has permitted systematizing Trefftz method and expanding its scope. The concept of TH-completeness has played a key role for such developments. This paper is devoted to revise the present state of these matters. Starting from the basic concepts of the algebraic theory, Green–Herrera formulas are presented and Localized Adjoint Method (LAM...

This paper is devoted to the error estimates for some weighted L projections. Nearly optimal estimates are obtained. These estimates can be applied to the analysis of the usual multigrid method, multilevel preconditioner and domain decomposition method for solving elliptic boundary problems whose coefficients have large jump discontinuities.

We consider the solution of the discrete linear system resulting from a mixed finite element discretization applied to a second-order elliptic boundary value problem in three dimensions. Based on a decomposition of the velocity space, these equations can be reduced to a discrete elliptic problem by eliminating the pressure through the use of substructures of the domain. The practicality of the ...

Domain decomposition methods used for solving linear systems are strongly related to dynamic substructuring methods commonly used to build a reduced model. In this study we investigate some theoretical relations between the methods. In particular we discuss the conceptual similarities between the Schur Complement solvers and the Craig-Bampton substructuring techniques, both for their primal and...

Poincaré type inequalities play a central role in the analysis of domain decomposition and multigrid methods for second-order elliptic problems. However, when the coefficient varies within a subdomain or within a coarse grid element, then standard condition number bounds for these methods may be overly pessimistic. In this short note we present new weighted Poincaré type inequalities for a clas...

We show that the decreasing dimension method proposed by Wang and Jiang in \Solution of the System of Linear Algebraic Equations by Decreasing Dimension", Appl. Math. Comput, 109 (2000) 51{57, is a type of Schur complement domain decomposition method. The decreasing dimension method is more expensive than the standard Schur complement domain decomposition method for solving any linear systems. ...

A conservative Galerkin domain decomposition method for time-dependent problems is given and analyzed. This method allows one to apply diierent domain decompositions at diierent time levels when necessary, in order to capture time-changing local phenomena, such as, propagating fronts or moving layers. Error estimates in the energy and L 2 norms are established. Numerical results are presented.

Abstract: We relate a particular version of a parallel multigrid method analyzed by C. Douglas, W. L. Miranker, and B. F. Smith ([4, 6]) to the domain decomposition method of D. Funaro, L. D. Marini, A. Quarteroni, and P. Zanolli ([7, 8]). We show that the parallel multigrid method is reducing computation to a small portion of the domain and then extending the solution to the entire domain usin...

We consider additive Schwarz methods for the biharmonic Dirichlet problem and show that the algorithms have optimal convergence properties for some conforming nite elements. Some multilevel methods are also discussed.