Substructuring domain decomposition scheme for unsteady problems

Domain decomposition methods are used for approximate solving boundary problems for partial differential equations on parallel computing systems. Specific features of unsteady problems are taken into account in the most complete way in iteration-free schemes of domain decomposition. Regionally-additive schemes are based on different classes of splitting schemes. In this paper we highlight a class of domain decomposition schemes which is based on the partition of the initial domain into subdomains with common boundary nodes. Using the partition of unit we have constructed and studied unconditionally stable schemes of domain decomposition based on two-component splitting: the problem within subdomain and the problem at their boundaries. As an example there is considered the Cauchy problem for evolutionary equations of first and second order with non-negative self-adjoint operator in a finite Hilbert space. The theoretical consideration is supplemented with numerical solving a model problem for the two-dimensional parabolic equation

A parallel space-time domain decomposition method for unsteady source inversion problems

In this paper, we propose a parallel space-time domain decomposition method for solving an unsteady source identification problem governed by the linear convection-diffusion equation. Traditional approaches require to solve repeatedly a forward parabolic system, an adjoint system and a system with respect to the unknowns. The three systems have to be solved one after another. These sequential steps are not desirable for large scale parallel computing. A space-time restrictive additive Schwarz method is proposed for a fully implicit space-time coupled discretization scheme to recover the time-dependent pollutant source intensity functions. We show with numerical experiments that the scheme works well with noise in the observation data. More importantly it is demonstrated that the parallel space-time Schwarz preconditioner is scalable on a supercomputer with over $10^3$ processors, thus promising for large scale applications

Overlapping Localized Exponential Time Differencing Methods for Diffusion Problems

The paper is concerned with overlapping domain decomposition and exponential time differencing for the diffusion equation discretized in space by cell-centered finite differences. Two localized exponential time differencing methods are proposed to solve the fully discrete problem: the first method is based on Schwarz iteration applied at each time step and involves solving stationary problems in the subdomains at each iteration, while the second method is based on the Schwarz waveform relaxation algorithm in which time-dependent subdomain problems are solved at each iteration. The convergence of the associated iterative solutions to the corresponding fully discrete multidomain solution and to the exact semi-discrete solution is rigorously proved. Numerical experiments are carried out to confirm theoretical results and to compare the performance of the two methods.Comment: 23 page

On Schwarz Methods for Nonsymmetric and Indefinite Problems

In this paper we introduce a new Schwarz framework and theory, based on the well-known idea of space decomposition, for nonsymmetric and indefinite linear systems arising from continuous and discontinuous Galerkin approximations of general nonsymmetric and indefinite elliptic partial differential equations. The proposed Schwarz framework and theory are presented in a variational setting in Banach spaces instead of Hilbert spaces which is the case for the well-known symmetric and positive definite (SPD) Schwarz framework and theory. Condition number estimates for the additive and hybrid Schwarz preconditioners are established. The main idea of our nonsymmetric and indefinite Schwarz framework and theory is to use weak coercivity (satisfied by the nonsymmetric and indefinite bilinear form) induced norms to replace the standard bilinear form induced norm in the SPD Schwarz framework and theory. Applications of the proposed nonsymmetric and indefinite Schwarz framework and theory. Applications of the proposed nonsymmetric and indefinite Schwarz framework to solutions of discontinuous Galerkin approximations of convection-diffusion problems are also discussed. Extensive 1-D numerical experiments are also provided to gauge the performance of the proposed Schwarz methods.Comment: 34 pages, 12 tables and 14 figure

Two-component domain decomposition scheme with overlapping subdomains for parabolic equations

An iteration-free method of domain decomposition is considered for approximate solving a boundary value problem for a second-order parabolic equation. A standard approach to constructing domain decomposition schemes is based on a partition of unity for the domain under the consideration. Here a new general approach is proposed for constructing domain decomposition schemes with overlapping subdomains based on indicator functions of subdomains. The basic peculiarity of this method is connected with a representation of the problem operator as the sum of two operators, which are constructed for two separate subdomains with the subtraction of the operator that is associated with the intersection of the subdomains. There is developed a two-component factorized scheme, which can be treated as a generalization of the standard Alternating Direction Implicit (ADI) schemes to the case of a special three-component splitting. There are obtained conditions for the unconditional stability of regionally additive schemes constructed using indicator functions of subdomains. Numerical results are presented for a model two-dimensional problem.Comment: 18 pages, 8 figure

V-cycle multigrid algorithms for discontinuous Galerkin methods on non-nested polytopic meshes

In this paper we analyse the convergence properties of V-cycle multigrid algorithms for the numerical solution of the linear system of equations arising from discontinuous Galerkin discretization of second-order elliptic partial differential equations on polytopal meshes. Here, the sequence of spaces that stands at the basis of the multigrid scheme is possibly non nested and is obtained based on employing agglomeration with possible edge/face coarsening. We prove that the method converges uniformly with respect to the granularity of the grid and the polynomial approximation degree p, provided that the number of smoothing steps, which depends on p, is chosen sufficiently large.Comment: 26 pages, 23 figures, submitted to Journal of Scientific Computin

Preconditioning the bidomain model with almost linear complexity

The bidomain model is widely used in electro-cardiology to simulate spreading of excitation in the myocardium and electrocardiograms. It consists of a system of two parabolic reaction diffusion equations coupled with an ODE system. Its discretisation displays an ill-conditioned system matrix to be inverted at each time step: simulations based on the bidomain model therefore are associated with high computational costs. In this paper we propose a preconditioning for the bidomain model either for an isolated heart or in an extended framework including a coupling with the surrounding tissues (the torso). The preconditioning is based on a formulation of the discrete problem that is shown to be symmetric positive semi-definite. A block $LU$ decomposition of the system together with a heuristic approximation (referred to as the monodomain approximation) are the key ingredients for the preconditioning definition. Numerical results are provided for two test cases: a 2D test case on a realistic slice of the thorax based on a segmented heart medical image geometry, a 3D test case involving a small cubic slab of tissue with orthotropic anisotropy. The analysis of the resulting computational cost (both in terms of CPU time and of iteration number) shows an almost linear complexity with the problem size, i.e. of type $n\log^\alpha(n)$ (for some constant $\alpha$) which is optimal complexity for such problems

Domain decomposition schemes for evolutionary equations of first order with not self-adjoint operators

Domain decomposition methods are essential in solving applied problems on parallel computer systems. For boundary value problems for evolutionary equations the implicit schemes are in common use to solve problems at a new time level employing iterative methods of domain decomposition. An alternative approach is based on constructing iteration-free methods based on special schemes of splitting into subdomains. Such regionally-additive schemes are constructed using the general theory of additive operator-difference schemes. There are employed the analogues of classical schemes of alternating direction method, locally one-dimensional schemes, factorization methods, vector and regularized additive schemes. The main results were obtained here for time-dependent problems with self-adjoint elliptic operators of second order. The paper discusses the Cauchy problem for the first order evolutionary equations with a nonnegative not self-adjoint operator in a finite-dimensional Hilbert space. Based on the partition of unit, we have constructed the operators of decomposition which preserve nonnegativity for the individual operator terms of splitting. Unconditionally stable additive schemes of domain decomposition were constructed using the regularization principle for operator-difference schemes. Vector additive schemes were considered, too. The results of our work are illustrated by a model problem for the two-dimensional parabolic equation

Robust multigrid for high-order discontinuous Galerkin methods: A fast Poisson solver suitable for high-aspect ratio Cartesian grids

We present a polynomial multigrid method for nodal interior penalty and local discontinuous Galerkin formulations of the Poisson equation on Cartesian grids. For smoothing we propose two classes of overlapping Schwarz methods. The first class comprises element-centered and the second face-centered methods. Within both classes we identify methods that achieve superior convergence rates, prove robust with respect to the mesh spacing and the polynomial order, at least up to ${P=32}$. Consequent structure exploitation yields a computational complexity of $O(PN)$, where $N$ is the number of unknowns. Further we demonstrate the suitability of the face-centered method for element aspect ratios up to 32

Parallel stochastic methods for PDE based grid generation

The efficient generation of meshes is an important step in the numerical solution of various problems in physics and engineering. We are interested in situations where global mesh quality and tight coupling to the physical solution is important. We consider elliptic PDE based mesh generation and present a method for the construction of adaptive meshes in two spatial dimensions using domain decomposition that is suitable for an implementation on parallel computing architectures. The method uses the stochastic representation of the exact solution of a linear mesh generator of Winslow type to find the points of the adaptive mesh along the subdomain interfaces. The meshes over the single subdomains can then be obtained completely independently of each other using the probabilistically computed solutions along the interfaces as boundary conditions for the linear mesh generator. Further to the previously acknowledged performance characteristics, we demonstrate how the stochastic domain decomposition approach is particularly suited to the problem of grid generation - generating quality meshes efficiently. In addition we show further improvements are possible using interpolation of the subdomain interfaces and smoothing of mesh candidates. An optimal placement strategy is introduced to automatically choose the number and placement of points along the interface using the mesh density function. Various examples of meshes constructed using this stochastic-deterministic domain decomposition technique are shown and compared to the respective single domain solutions using a representative mesh quality measure. A brief performance study is included to show the viability of the stochastic domain decomposition approach and to illustrate the effect of algorithmic choices on the solver's efficiency.Comment: 25 pages, 11 figure