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In this paper, we propose a two-level additive Schwarz domain decomposition preconditioner for the symmetric interior penalty Galerkin method for a second-order elliptic boundary value problem with highly heterogeneous coefficients. A specific feature of this preconditioner is that it is based on adaptively enriching its coarse space with functions created by solving generalized eigenvalue problems on thin patches covering the subdomain interfaces. It is shown that the condition number of the underlined preconditioned system is independent of the contrast if an adequate number of functions are used to enrich the coarse space. Numerical results are provided to confirm this claim.

We propose two variants of the overlapping additive Schwarz method for the finite element discretiza- tion of the elliptic problem in 3D with highly heterogeneous coefficients. The methods are efficient and simple to construct using the abstract framework of the additive Schwarz method, and an idea of adaptive coarse spaces. In one variant, the coarse space consists of finite element functions associated with the wire basket nodes and functions based on solving some generalized eigenvalue problem on the faces, and in the other variant, it contains functions associated with the vertex nodes with functions based on solving some generalized eigenvalue problems on subdomain faces and on subdomain edges. The functions that are used to build the coarse spaces are chosen adaptively, they correspond to the eigenvalues that are smaller than a given threshold. The convergence rate of the preconditioned conjugate gradients method in both cases, is shown to be independent of the variations in the coefficients for sufficient number of eigenfunctions in the coarse space. Numerical results are given to support the theory.

In many real physical phenomena, there is heterogeneity, e.g., in some ground flow problems in heterogeneous media.When some finite element discretization method is applied to a physical model, one usually obtains a discrete problem which is very hard to solve by a preconditioned iterative method like, e.g., Preconditioned Conjugate Gradient (PCG) method.

In this paper, a domain decomposition parallel preconditioner for the 4th order multiscale elliptic problem in 2D with highly heterogeneous coefficients is constructed. The problem is discretized by the conforming \(C^1\) reduced Hsieh-Tough-Tocher (HCT) macro element. The proposed preconditioner is based on the classical overlapping Schwarz method and is constructed using an abstract framework of the Additive Schwarz Method. The coarse space consists of multiscale finite element functions associated with the edges and is enriched with functions based on solving carefully constructed generalized eigenvalue problems locally on each edge. The convergence rate of the Preconditioned Conjugate Method of our method is independent of the variations in the coefficients for a sufficient number of eigenfunctions in the coarse space.

The SHEM (Spectral Harmonically Enriched Multiscale) coarse space is a new coarse space for arbitrary overlapping or non-overlapping domain decomposition methods.

In this paper an overlapping additive Schwarz method with a spectrally enriched coarse space is proposed. The method is for solving the standard Finite Element discretization of second order elliptic problems in there dimensions with discontinuous coefficients, where the discontinuities are inside subdomains and across subdomain boundaries. In case when the coarse space is large enough the convergence of the PCG method is independent of jumps in the coefficient.

A model combining the first-order and the second-order variational regularizations for the purpose of 3D surface reconstruction based on 2D sparse data is proposed. The model includes a hybrid fidelity constraint which allows the initial conditions to be switched flexibly between vectors and elevations. A numerical algorithm based on the augmented Lagrangian method is also proposed. The numerical experiments are presented, showing its excellent performance both in designing cartoon characters, as well as in recovering oriented three dimensional maps from contours or points with elevation information. acceptedVersion

A symmetric and a nonsymmetric variant of the additive Schwarz preconditioner are proposed for the solution of a class of finite volume element discretization of the symmetric elliptic problem in two dimensions, with large jumps in the entries of the coefficient matrices across subdomains. It is shown that the convergence of the preconditioned generalized minimal residual iteration using the proposed preconditioners depends polylogarithmically, in other words weakly, on the mesh parameters, and that they are robust with respect to the jumps in the coefficients. publishedVersion

We present an analysis of the additive average Schwarz preconditioner with two newly proposed adaptively enriched coarse spaces which was presented at the 23rd International conference on domain decomposition methods in Korea, for solving second order elliptic problems with highly varying and discontinuous coefficients. It is shown that the condition number of the preconditioned system is bounded independently of the variations and the jumps in the coefficient, and depends linearly on the mesh parameter ratio H/h, that is the ratio between the subdomain size and the mesh size, thereby retaining the same optimality and scalablity of the original additive average Schwarz preconditioner., Comment: Corrected typos, improved numerical section, figures replaced

In domain decomposition we decompose a global domain into subdomains. The shape and form of these subdomains may have an effect upon the overall performance of our methods. In this paper we study the effects of irregular subdomain boundaries on some domain decomposition algorithms. We reintroduce the additive average Schwarz method. This method is very flexible with respect to subdomain shapes, and we make the claim that for the scalar elliptic equation, the established convergences estimate for this method also hold under the weak conditions that subdomains are John domains with a uniformly bounded John constant. We present some numerical results to show how various subdomain shapes and geometries can effect the condition number of the preconditioned system.

In this paper we introduce an Average Additive Schwarz method with spectrally enriched coarse grids for a standard Finite Element discretization of a second order elliptic problem with discontinuous coefficients, where the discontinuities are inside subdomains and across subdomain boundaries. The proposed methods depend linearly on the mesh parameters H∕h, i.e., depending on the distribution of the coefficient in the model problem, the parameters describing the convergence of the PCG method used to solve the preconditioned system depends linearly or on the mesh parameters. In case when the coarse space is large enough the convergence of the PCG method is independent of jumps in the coefficient.

In this note we propose a local post-refinement technique, which can be used to provide the overall conformity of tetrahedral and hexahedral meshes meeting at the planar interface, which presents a quite common situation in many simulations of real-life problems. The same technique can be also used for the case of two adjacent non-matching hexahedral meshes.

In this paper, we present two variants of the Additive Schwarz Method (ASM) for a Crouzeix-Raviart finite volume (CRFV) discretization of the second order elliptic problem with discontinuous coefficients, where the discontinuities are only across subdomain boundaries. The resulting system, which is nonsymmetric, is solved using the preconditioned GMRES iteration, where in one variant of the ASM the preconditioner is symmetric while in the other variant it is nonsymmetric. The proposed methods are almost optimal, in the sense that the convergence of the GMRES iteration, in the both cases, depend only poly-logarithmically on the mesh parameters.

Image denoising is one of the fundamental problems in the image processing. In a PDE based approach for image processing, the simplest possible method for denoising is to solve the heat equation. However such a diffusion equation will destroy sharp edges in the image. An approach known for preserving the edges while denoising is called the classical Rudin-Osher-Fatemi (ROF) method based on the total variation (TV) regularization. Recently, an algorithm, also known as the TV-Stokes, based on two minimization steps involving the smoothing of the tangential field and then the reconstruction of the image has been proposed. The latter produces images without the blocky effect which we observe in the case of the ROF model. An iterative regularization method for the total variation based image restoration has recently been proposed giving significant improvement over the classical method in the quality of the restored image. In this paper we propose a similar algorithm for the TV-Stokes denoising algorithm.