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Analysis of patch substructuring methods

Martin Gander, Laurence Halpern, Frédéric Magoulès, Francois Roux (2007)

International Journal of Applied Mathematics and Computer Science

Patch substructuring methods are non-overlapping domain decomposition methods like classical substructuring methods, but they use information from geometric patches reaching into neighboring subdomains condensated, on the interfaces to enhance the performance of the method, while keeping it non-overlapping. These methods are very convenient to use in practice, but their convergence properties have not been studied yet. We analyze geometric patch substructuring methods for the special case of one...

Analysis of the FEM and DGM for an elliptic problem with a nonlinear Newton boundary condition

Feistauer, Miloslav, Bartoš, Ondřej, Roskovec, Filip, Sändig, Anna-Margarete (2017)

Proceedings of Equadiff 14

The paper is concerned with the numerical analysis of an elliptic equation in a polygon with a nonlinear Newton boundary condition, discretized by the finite element or discontinuous Galerkin methods. Using the monotone operator theory, it is possible to prove the existence and uniqueness of the exact weak solution and the approximate solution. The main attention is paid to the study of error estimates. To this end, the regularity of the weak solution is investigated and it is shown that due to...

Analysis of two-level domain decomposition preconditioners based on aggregation

Marzio Sala (2004)

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

In this paper we present two-level overlapping domain decomposition preconditioners for the finite-element discretisation of elliptic problems in two and three dimensions. The computational domain is partitioned into overlapping subdomains, and a coarse space correction is added. We present an algebraic way to define the coarse space, based on the concept of aggregation. This employs a (smoothed) aggregation technique and does not require the introduction of a coarse grid. We consider a set of assumptions...

Analysis of two-level domain decomposition preconditioners based on aggregation

Marzio Sala (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper we present two-level overlapping domain decomposition preconditioners for the finite-element discretisation of elliptic problems in two and three dimensions. The computational domain is partitioned into overlapping subdomains, and a coarse space correction is added. We present an algebraic way to define the coarse space, based on the concept of aggregation. This employs a (smoothed) aggregation technique and does not require the introduction of a coarse grid. We consider a...

Anisotropic h p -adaptive method based on interpolation error estimates in the H 1 -seminorm

Vít Dolejší (2015)

Applications of Mathematics

We develop a new technique which, for the given smooth function, generates the anisotropic triangular grid and the corresponding polynomial approximation degrees based on the minimization of the interpolation error in the broken H 1 -seminorm. This technique can be employed for the numerical solution of boundary value problems with the aid of finite element methods. We present the theoretical background of this approach and show several numerical examples demonstrating the efficiency of the proposed...

Anisotropic interpolation error estimates via orthogonal expansions

Mingxia Li, Shipeng Mao (2013)

Open Mathematics

We prove anisotropic interpolation error estimates for quadrilateral and hexahedral elements with all possible shape function spaces, which cover the intermediate families, tensor product families and serendipity families. Moreover, we show that the anisotropic interpolation error estimates hold for derivatives of any order. This goal is accomplished by investigating an interpolation defined via orthogonal expansions.

Anisotropic mesh adaption: application to computational fluid dynamics

Simona Perotto (2005)

Bollettino dell'Unione Matematica Italiana

In this communication we focus on goal-oriented anisotropic adaption techniques. Starting point has been the derivation of suitable anisotropic interpolation error estimates for piecewise linear finite elements, on triangular grids in 2 ⁢ D . Then we have merged these interpolation estimates with the dual-based a posteriori error analysis proposed by R. Rannacher and R. Becker. As examples of this general anisotropic a posteriori analysis, elliptic, advection-diffusion-reaction and the Stokes problems...

Anisotropic mesh refinement in polyhedral domains: error estimates with data in L2(Ω)

Thomas Apel, Ariel L. Lombardi, Max Winkler (2014)

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

The paper is concerned with the finite element solution of the Poisson equation with homogeneous Dirichlet boundary condition in a three-dimensional domain. Anisotropic, graded meshes from a former paper are reused for dealing with the singular behaviour of the solution in the vicinity of the non-smooth parts of the boundary. The discretization error is analyzed for the piecewise linear approximation in the H1(Ω)- and L2(Ω)-norms by using a new quasi-interpolation operator. This new interpolant...

Application of relaxation scheme to degenerate variational inequalities

Jela Babušíková (2001)

Applications of Mathematics

In this paper we are concerned with the solution of degenerate variational inequalities. To solve this problem numerically, we propose a numerical scheme which is based on the relaxation scheme using non-standard time discretization. The approximate solution on each time level is obtained in the iterative way by solving the corresponding elliptic variational inequalities. The convergence of the method is proved.

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