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Nearly optimal convergence result for multigrid with aggressive coarsening and polynomial smoothing

Petr Vaněk, Marian Brezina (2013)

Applications of Mathematics

We analyze a general multigrid method with aggressive coarsening and polynomial smoothing. We use a special polynomial smoother that originates in the context of the smoothed aggregation method. Assuming the degree of the smoothing polynomial is, on each level k , at least C h k + 1 / h k , we prove a convergence result independent of h k + 1 / h k . The suggested smoother is cheaper than the overlapping Schwarz method that allows to prove the same result. Moreover, unlike in the case of the overlapping Schwarz method, analysis...

Nonconforming Galerkin methods based on quadrilateral elements for second order elliptic problems

Jim Douglas Jr., Juan E. Santos, Dongwoo Sheen, Xiu Ye (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

Low-order nonconforming Galerkin methods will be analyzed for second-order elliptic equations subjected to Robin, Dirichlet, or Neumann boundary conditions. Both simplicial and rectangular elements will be considered in two and three dimensions. The simplicial elements will be based on P1, as for conforming elements; however, it is necessary to introduce new elements in the rectangular case. Optimal order error estimates are demonstrated in all cases with respect to a broken norm in H1(Ω)...

Numerical study of two sparse AMG-methods

Janne Martikainen (2003)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

A sparse algebraic multigrid method is studied as a cheap and accurate way to compute approximations of Schur complements of matrices arising from the discretization of some symmetric and positive definite partial differential operators. The construction of such a multigrid is discussed and numerical experiments are used to verify the properties of the method.

Numerical Study of Two Sparse AMG-methods

Janne Martikainen (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

A sparse algebraic multigrid method is studied as a cheap and accurate way to compute approximations of Schur complements of matrices arising from the discretization of some symmetric and positive definite partial differential operators. The construction of such a multigrid is discussed and numerical experiments are used to verify the properties of the method.

Object oriented design philosophy for scientific computing

Philippe R. B. Devloo, Gustavo C. Longhin (2002)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

This contribution gives an overview of current research in applying object oriented programming to scientific computing at the computational mechanics laboratory (LABMEC) at the school of civil engineering – UNICAMP. The main goal of applying object oriented programming to scientific computing is to implement increasingly complex algorithms in a structured manner and to hide the complexity behind a simple user interface. The following areas are current topics of research and documented within the...

Object oriented design philosophy for scientific computing

Philippe R.B. Devloo, Gustavo C. Longhin (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

This contribution gives an overview of current research in applying object oriented programming to scientific computing at the computational mechanics laboratory (LABMEC) at the school of civil engineering – UNICAMP. The main goal of applying object oriented programming to scientific computing is to implement increasingly complex algorithms in a structured manner and to hide the complexity behind a simple user interface. The following areas are current topics of research and documented within the...

On a Parallel Implementation of the Mortar Element Method

Gassav S. Abdoulaev, Yves Achdou, Yuri A. Kuznetsov, Christophe Prud'homme (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We discuss a parallel implementation of the domain decomposition method based on the macro-hybrid formulation of a second order elliptic equation and on an approximation by the mortar element method. The discretization leads to an algebraic saddle- point problem. An iterative method with a block- diagonal preconditioner is used for solving the saddle- point problem. A parallel implementation of the method is emphasized. Finally the results of numerical experiments are presented.

On adaptive BDDC for the flow in heterogeneous porous media

Bedřich Sousedík (2019)

Applications of Mathematics

We study a method based on Balancing Domain Decomposition by Constraints (BDDC) for numerical solution of a single-phase flow in heterogeneous porous media. The method solves for both flux and pressure variables. The fluxes are resolved in three steps: the coarse solve is followed by subdomain solves and last we look for a divergence-free flux correction and pressures using conjugate gradients with the BDDC preconditioner. Our main contribution is an application of the adaptive algorithm for selection...

On monotone and Schwarz alternating methods for nonlinear elliptic PDEs

Shiu-Hong Lui (2001)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

The Schwarz alternating method can be used to solve elliptic boundary value problems on domains which consist of two or more overlapping subdomains. The solution is approximated by an infinite sequence of functions which results from solving a sequence of elliptic boundary value problems in each of the subdomains. In this paper, proofs of convergence of some Schwarz alternating methods for nonlinear elliptic problems which are known to have solutions by the monotone method (also known as the method...

Currently displaying 141 – 160 of 221