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An earlier paper proved the convergence of a variable stepsize Bregman operator splitting algorithm (BOSVS) for minimizing ø(Bu) + H(u), where H and ø are convex functions, and ø is possibly nonsmooth. The algorithm was shown to be relatively efficient when applied to partially parallel magnetic resonance image reconstruction problems. In this paper, the convergence rate of BOSVS is analyzed. When H(u) = ‖Au - f ‖2, where A is a matrix, it is shown that for an ergodic approximation uk obtained by averaging k BOSVS iterates, the error in the objective value ø(Buk)+H(uk) is O(1/k). When the optimization problem has a unique solution u*, we obtain the estimate ‖uk-u*‖ = O(1/ √ k). The theoretical analysis is compared to observed convergence rates for partially parallel magnetic resonance image reconstruction problems where A is a large dense ill-conditioned matrix. [ABSTRACT FROM AUTHOR]

Local mesh-refining algorithms known from adaptive finite element methods are adopted for locally conservative and monotone finite volume discretizations of boundary value problems for steady-state convection-diffusion-reaction equations. The paper establishes residual-type explicit error estimators and averaging techniques for a posteriori finite volume error control with and without upwind in global H1- and L2-norms. Reliability and efficiency are verified theoretically and confirmed empirically with experimental support for the superiority of the suggested adaptive mesh-refining algorithms over uniform mesh refining. A discussion of adaptive computations in the simulation of contaminant concentration in a nonhomogeneous water reservoir concludes the paper. [ABSTRACT FROM AUTHOR]

Discontinuous Galerkin (DG) and mixed finite element (MFE) methods are two popular methods that possess local mass conservation. In this paper we investigate DG-DG and DG-MFE domain decomposition couplings using mortar finite elements to impose weak continuity of fluxes and pressures on the interface. The subdomain grids need not match, and the mortar grid may be much coarser, giving a two-scale method. Convergence results in terms of the fine subdomain scale h and the coarse mortar scale H are established for both types of couplings. In addition, a nonoverlapping parallel domain decomposition algorithm is developed, which reduces the coupled system to an interface mortar problem. The properties of the interface operator are analyzed. [ABSTRACT FROM AUTHOR]

We consider two-dimensional elliptic problems with discontinuous coefficients discretized by the finite element method on geometrically conforming nonmatching triangulations across the interface using the mortar technique. The resulting discrete problem is solved by a dual-primal FETI method. In this paper we introduce and analyze a preconditioner with a special scaling of coefficients and step parameters and establish convergence bounds. We show that the preconditioner is almost optimal with constants independent of the jumps of coefficients and step parameters. Extensive computational evidence is presented that illustrates an almost optimal convergence for a variety of situations (distribution of subregions, grid assignment, grid ratios, number of subregions) for both continuous and discontinuous problems. [ABSTRACT FROM AUTHOR]

We present and study a nonconforming domain decomposition method for the discretization of the three-dimensional Stokes problem in the velocity-pressure formulation. The approximation is based on some local mixed finite elements for nonmatching tetrahedral grids. The aim pursued is a systematic construction of the mortared discrete velocity space, the pressure being not subjected to any matching constraints across the interfaces. Using the bubble stabilization techniques, applied in Brezzi and Marini's paper to the three fields method [Math. Cornp., 70 (2001), pp. 911-934], allows us to define an algorithm which is easy to implement. The numerical analysis relies on the pressure-splitting argument of Boland and Nicolaides and allows us to establish an inf-sup condition with a constant that does not depend on the mesh size or on the total number of the subdomains. Then, by the Berger-Scott-Strang lemma written down for our saddle point system we derive optimal accuracy results. [ABSTRACT FROM AUTHOR]

In this paper, we propose a nonoverlapping domain decomposition method for solving the three-dimensional Maxwell equations, based on the edge element discretization. For the Schur complement system on the interface, we construct an efficient preconditioner by introducing two special coarse subspaces defined on the nonoverlapping sub domains. It is shown that the condition number of the preconditioned system grows only polylogarithmically with the ratio between the subdomain diameter and the finite element mesh size but possibly depends on the jumps of the coefficients. [ABSTRACT FROM AUTHOR]

In the first part of the paper, we introduce an overlapping mortar finite element method for solving two-dimensional elliptic problems discretized on overlapping nonmatching grids. We prove an optimal error bound and estimate the condition numbers of certain overlapping Schwarz preconditioned systems for the two-subdomain case. We show that the error bound is independent of the size of the overlap and the ratio of the mesh parameters. In the second part, we introduce three additive Schwarz preconditioned conjugate gradient algorithms based on the trivial and harmonic extensions. We provide estimates for the spectral bounds on the condition numbers of the preconditioned operators. We show that although the error bound is independent of the size of the overlap, the condition number does depend on it. Numerical examples are presented to support our theory. [ABSTRACT FROM AUTHOR]

The mortar methods are based on domain decomposition and they allow for the coupling of different variational approximations in different subdomains. The resulting methods are nonconforming but still yield optimal approximations. In this paper, we will discuss iterative substructuring algorithms for the algebraic systems arising from the discretization of symmetric, second-order, elliptic equations in two dimensions. Both spectral and finite element methods, for geometrically conforming as well as nonconforming domain decompositions, are studied. In each case, we obtain a polylogarithmic bound on the condition number of the preconditioned matrix. [ABSTRACT FROM AUTHOR]

In this paper, we study the additive Schwarz method for the h-p version of the finite element method in three dimensions. The main idea is to treat separately the h-version (linear) components and the p-version (high-order) components by a vertex-based method. It can also be viewed as a three-level method with the level being the linear finite element approximation on the coarse mesh, the linear finite element approximation on the fine mesh, and the high-order finite element approximation on the fine mesh, respectively. The resulting algorithm can be implemented in parallel on the subdomain level for the h-version components and on the element level for the p-version components. The condition number is of order [This symbol cannot be presented in ASCII format](1 + ln Hipi/hi)⇔2, where Hi stands for the diameter of the subdomain Ωi, hi is the diameter of the elements in Ωi, and pi is the maximum of the polynomial degrees used in Ωi. [ABSTRACT FROM AUTHOR]