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Implementation of optimal Galerkin and Collocation approximations of PDEs with Random Coefficients⋆⋆⋆

J. Beck, F. Nobile, L. Tamellini, R. Tempone (2011)

ESAIM: Proceedings

In this work we first focus on the Stochastic Galerkin approximation of the solution u of an elliptic stochastic PDE. We rely on sharp estimates for the decay of the coefficients of the spectral expansion of u on orthogonal polynomials to build a sequence of polynomial subspaces that features better convergence properties compared to standard polynomial subspaces such as Total Degree or Tensor Product. We consider then the Stochastic Collocation method, and use the previous estimates to introduce...

Implicit a posteriori error estimation using patch recovery techniques

Tamás Horváth, Ferenc Izsák (2012)

Open Mathematics

We develop implicit a posteriori error estimators for elliptic boundary value problems. Local problems are formulated for the error and the corresponding Neumann type boundary conditions are approximated using a new family of gradient averaging procedures. Convergence properties of the implicit error estimator are discussed independently of residual type error estimators, and this gives a freedom in the choice of boundary conditions. General assumptions are elaborated for the gradient averaging...

Improved convergence bounds for smoothed aggregation method: linear dependence of the convergence rate on the number of levels

Jan Brousek, Pavla Fraňková, Petr Vaněk (2016)

Czechoslovak Mathematical Journal

The smoothed aggregation method has became a widely used tool for solving the linear systems arising by the discretization of elliptic partial differential equations and their singular perturbations. The smoothed aggregation method is an algebraic multigrid technique where the prolongators are constructed in two steps. First, the tentative prolongator is constructed by the aggregation (or, the generalized aggregation) method. Then, the range of the tentative prolongator is smoothed by a sparse linear...

Inf-sup stable nonconforming finite elements of higher order on quadrilaterals and hexahedra

Gunar Matthies (2007)

ESAIM: Mathematical Modelling and Numerical Analysis

We present families of scalar nonconforming finite elements of arbitrary order r ≥ 1 with optimal approximation properties on quadrilaterals and hexahedra. Their vector-valued versions together with a discontinuous pressure approximation of order r - 1 form inf-sup stable finite element pairs of order r for the Stokes problem. The well-known elements by Rannacher and Turek are recovered in the case r=1. A numerical comparison between conforming and nonconforming discretisations will be given. Since higher order...

Inverted finite elements : a new method for solving elliptic problems in unbounded domains

Tahar Zamène Boulmezaoud (2005)

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

In this paper, we propose a new numerical method for solving elliptic equations in unbounded regions of ℝ n . The method is based on the mapping of a part of the domain into a bounded region. An appropriate family of weighted spaces is used for describing the growth or the decay of functions at large distances. After exposing the main ideas of the method, we analyse carefully its convergence. Some 3D computational results are displayed to demonstrate its efficiency and its high performance.

Inverted finite elements: a new method for solving elliptic problems in unbounded domains

Tahar Zamène Boulmezaoud (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper, we propose a new numerical method for solving elliptic equations in unbounded regions of ℝ n . The method is based on the mapping of a part of the domain into a bounded region. An appropriate family of weighted spaces is used for describing the growth or the decay of functions at large distances. After exposing the main ideas of the method, we analyse carefully its convergence. Some 3D computational results are displayed to demonstrate its efficiency and its high performance.

Iterative schemes for high order compact discretizations to the exterior Helmholtz equation∗

Yogi Erlangga, Eli Turkel (2012)

ESAIM: Mathematical Modelling and Numerical Analysis

We consider high order finite difference approximations to the Helmholtz equation in an exterior domain. We include a simplified absorbing boundary condition to approximate the Sommerfeld radiation condition. This yields a large, but sparse, complex system, which is not self-adjoint and not positive definite. We discretize the equation with a compact fourth or sixth order accurate scheme. We solve this large system of linear equations with a Krylov subspace iterative method. Since the method converges...

Iterative schemes for high order compact discretizations to the exterior Helmholtz equation∗

Yogi Erlangga, Eli Turkel (2012)

ESAIM: Mathematical Modelling and Numerical Analysis

We consider high order finite difference approximations to the Helmholtz equation in an exterior domain. We include a simplified absorbing boundary condition to approximate the Sommerfeld radiation condition. This yields a large, but sparse, complex system, which is not self-adjoint and not positive definite. We discretize the equation with a compact fourth or sixth order accurate scheme. We solve this large system of linear equations with a Krylov subspace iterative method. Since the method converges...

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