A general framework for substructuring-based domain decomposition methods for models having nonlocal interactions [including apppendix B

A rigorous mathematical framework is provided for a substructuring-based domain-decomposition approach for nonlocal problems that feature interactions between points separated by a finite distance. Here, by substructuring it is meant that a traditional geometric configuration for local partial differential equation problems is used in which a computational domain is subdivided into non-overlapping subdomains. In the nonlocal setting, this approach is substructuring-based in the sense that those subdomains interact with neighboring domains over interface regions having finite volume, in contrast to the local PDE setting in which interfaces are lower dimensional manifolds separating abutting subdomains Key results include the equivalence between the global, single-domain nonlocal problem and its multi-domain reformulation, both at the continuous and discrete levels. These results provide the rigorous foundation necessary for the development of efficient solution strategies for nonlocal domain-decomposition methods.

A mathematical framework is provided for a substructuring‐based domain decomposition (DD) approach for nonlocal problems that features interactions between points separated by a finite distance. Here, by substructuring it is meant that a traditional geometric configuration for local partial differential equation (PDE) problems is used in which a computational domain is subdivided into non‐overlapping subdomains. In the nonlocal setting, this approach is substructuring‐based in the sense that those subdomains interact with neighboring domains over interface regions having finite volume, in contrast to the local PDE setting in which interfaces are lower dimensional manifolds separating abutting subdomains. Key results include the equivalence between the global, single‐domain nonlocal problem and its multi‐domain reformulation, both at the continuous and discrete levels. These results provide the rigorous foundation necessary for the development of efficient solution strategies for nonlocal DD methods.

Computation of two-phase flow in steam generator using Domain Decomposition and Local Zoom methods

We consider the algebraic and geometric issues of the advanced parallel domain decomposition methods (DDMs) for solving very large non-symmetric systems of linear algebraic equations (SLAEs) that arise in the finite volume or the finite element approximation of the multi-dimensional boundary value problems on the non-structured grids. The main approaches in question for DDM include the balancing decomposition of the grid computational domain into parameterized overlapping or non-overlapping subdomains with different interface conditions on the internal boundaries. Also, we use two different sructures of the contacting the neigbour grid subdomains: with definition or without definition of the node dividers (separators) as the special grid subdomain. The proposed Schwarz parallel additive algorithms are based on the “total-flexible” multi-preconditioned semi-conjugate direction methods in the Krylov block subspaces. The acceleration of two-level iterative processes is provided by means of aggregation, or coarse grid correction, with different orders of basic functions, which realize a low - rank approximation of the original matrix. The auxiliary subsystems in subdomains are solved by direct or by the Krylov iterative methods. The parallel implementation of algorithms is based on hybrid programming with MPI-processes and multi-thread computing for the upper and the low levels of iterations, respectively. We describe some characteristic features of the computational technologies of DDMs that are realized within the framework of the library KRYLOV in the Institute of Computational Mathematics and Mathematical Geophysics, SB RAS, Novosibirsk. The technical requirements for this code are based on the absence of the program constraints on the degree of freedom and on the number of processor units. The conceptions of the creating the unified numerical envirement for DDMs are presented and discussed.

In the paper, we develop and analyze a new mass-preserving splitting domain decomposition method over multiple sub-domains for solving parabolic equations. The domain is divided into non-overlapping multi-bock sub-domains. On the interfaces of sub-domains, the interface fluxes are computed by the semi-implicit (explicit) flux scheme. The solutions and fluxes in the interiors of sub-domains are computed by the splitting one-dimensional implicit solution-flux coupled scheme. The important feature is that the proposed scheme is mass conservative over multiple non-overlapping sub-domains. Analyzing the mass-preserving S-DDM scheme is difficult over non-overlapping multi-block sub-domains due to the combination of the splitting technique and the domain decomposition at each time step. We prove theoretically that our scheme satisfies conservation of mass over multi-block non-overlapping sub-domains and it is unconditionally stable. We further prove the convergence and obtain the error estimate in L2-norm. Numerical experiments confirm theoretical results.

A Domain Decomposition Method for the Exterior Helmholtz Problem

This article presents a finite-element-based domain decomposition method (DDM) for thermal analysis of 3-D large-scale and geometrically complicated integrated systems. The presented DDM decomposes the entire problem domain into many nonoverlapping subdomains, and a novel transmission condition (TC) based on a fictitious thermal contact resistance is introduced and then imposed through an interior penalty (IP) Galerkin procedure to enforce continuity at the interfaces between subdomains. The major advantage of the novel TC combined with the IP Galerkin procedure is that no extra unknowns are introduced and the system matrix arising from the finite element method will be symmetric and positive definite. Furthermore, the solving procedure of matrix equation adopts the preconditioned conjugate gradient method with a two-level preconditioner. To verify the convergence and accuracy, this article calculates the transient temperature profile of a brick and compares the computed numerical approximation with the exact solution. Meanwhile, the solving advantage of the proposed DDM is shown by comparing with Robin-TC DDM. Finally, numerical results for a 3-D integration package are shown to demonstrate the accuracy and superior performance of the proposed DDM.

The first domain decomposition methods for partial differential equations were already developed in 1870 by H. A. Schwarz. Here we consider a nonlocal Dirichlet problem with variable coefficients, where a nonlocal diffusion operator is used. We find that domain decomposition methods like the so-called Schwarz methods seem to be a natural way to solve these nonlocal problems. In this work, we show the convergence for nonlocal problems, where specific symmetric kernels are employed, and present the implementation of the multiplicative and additive Schwarz algorithms in the above-mentioned nonlocal setting.

A discrete sensitivity analysis algorithm had previously been developed and applied to two-dimensional aerodynamic optimization problems, where the computational domains were discretized by using single grids. The sparse, unsymmetric systems of linear equations resulting from this algorithm were solved by a direct matrix inversion matrix. However, for large two-dimensional problems and, practically, all three-dimensional problems, direct inversion methods become inapplicable, primarily due to the prohibitive computer storage needed. In an attempt to alleviate such hindrances, the sensitivity analysis with domain decomposition (SADD) scheme was developed. This scheme divides the computational domain into smaller and nonoverlapping subdomains (multiblock grids) that are solved separately. Then, the final solution is constructed from the subdomain solutions. As the number of grid points in the interface boundaries of the subdomains becomes large, the computer memory required to store the effective coefficient matrix of these interface points starts to increase. Presented in this Technical Note is a preconditioned iterative procedure to overcome this particular problem.

A hybrid approach for the solution of linear elliptic PDEs, based on the unified transform method in conjunction with domain decomposition techniques, is introduced. Given a well-posed boundary value problem, the proposed methodology relies on the derivation of an approximate global relation, which is an equation that couples the finite Fourier transforms of all the boundary values. The computational domain is hierarchically decomposed into several nonoverlapping subdomains; for each of those subdomains, a unique approximate global relation is derived. Then, by introducing a modified Dirichlet-to-Neumann iterative algorithm, it is possible to compute the solution and its normal derivative at the resulting interfaces. By considering several hierarchical levels, higher spatial resolution can be achieved. There are three main advantages associated with the proposed approach. First, since the unified transform is a boundary-based technique, the interior of each subdomain does not need to be discretized; thus, no mesh generation is required. Additionally, the Dirichlet and Neumann values can be computed on the interfaces with high accuracy, using a collocation technique in the complex Fourier plane. Finally, the interface values at each hierarchical level can be computed in parallel by considering a quadtree decomposition in conjunction with the iterative Dirichlet-to-Neumann algorithm. The proposed methodology is analysed both regarding implementation details and computational complexity. Moreover, numerical results are presented, assessing the performance of the solver.

The first part of the thesis at hand deals with a finite volume method for time-dependent advection-diffusion-reaction equations. By using dual control volumes based on a common finite element triangulation, our discretisation can be formulated as a (conforming) generalised Galerkin method. We extend well-known convergence results to time-dependent problems, where advection dominated cases are taken into account by an upwind modification of the method.In the second part of this thesis we develop a new domain decomposition method for the parabolic problems that we looked at in the first part. It concerns an iteration-by-subdomain method of Dirichlet-Robin type with non-overlapping subdomains. As our domain decomposition algorithm aims for the direct application to parabolic problems without preceding discretisation in time, we must construct specific Steklov-Poincaré operators, and end up with a method of waveform relaxation type. Linear convergence of the method is shown on the continuous level as well as in the semidiscrete case, where the afore examined finite volume discretisation is applied. We state an optimisation strategy for the transmission conditions at the interface that improves the efficiency considerably. Finally we illustrate our theoretical conclusions by numerical results.

We prove existence, qualitative properties and asymptotic behavior of positive solutions to the doubly critical problem $$\begin{aligned} (-\Delta )^s u=\vartheta \frac{u}{|x|^{2s}}+u^{2_s^*-1}, \quad u\in \dot{H}^s(\mathbb {R}^N), \quad N> 2s,\quad 0<s<1. \end{aligned}$$ The technique that we use to prove the existence is based on variational arguments. The qualitative properties are obtained by using the moving plane method, in a nonlocal setting, on the whole \(\mathbb {R}^{N}\) and some comparison results. Moreover, in order to find the asymptotic behavior of solutions, we use a representation result that allows to transform the original problem into a different nonlocal problem in a weighted fractional space.

In this paper we consider 2D nonlocal diffusion models with a finite nonlocal horizon parameterδcharacterizing the range of nonlocal interactions, and consider the treatment of Neumann-like boundary conditions that have proven challenging for discretizations of nonlocal models. We propose a new generalization of classical local Neumann conditions by converting the local flux to a correction term in the nonlocal model, which provides an estimate for the nonlocal interactions of each point with points outside the domain. While existing 2D nonlocal flux boundary conditions have been shown to exhibit at most first order convergence to the local counter part asδ → 0, the proposed Neumann-type boundary formulation recovers the local case asO(δ2) in theL∞(Ω) norm, which is optimal considering theO(δ2) convergence of the nonlocal equation to its local limit away from the boundary. We analyze the application of this new boundary treatment to the nonlocal diffusion problem, and present conditions under which the solution of the nonlocal boundary value problem converges to the solution of the corresponding local Neumann problem as the horizon is reduced. To demonstrate the applicability of this nonlocal flux boundary condition to more complicated scenarios, we extend the approach to less regular domains, numerically verifying that we preserve second-order convergence for non-convex domains with corners. Based on the new formulation for nonlocal boundary condition, we develop an asymptotically compatible meshfree discretization, obtaining a solution to the nonlocal diffusion equation with mixed boundary conditions that converges withO(δ2) convergence.

In this paper we consider 2D nonlocal diffusion models with a finite nonlocal horizon parameterδcharacterizing the range of nonlocal interactions, and consider the treatment of Neumann-like boundary conditions that have proven challenging for discretizations of nonlocal models. We propose a new generalization of classical local Neumann conditions by converting the local flux to a correction term in the nonlocal model, which provides an estimate for the nonlocal interactions of each point with points outside the domain. While existing 2D nonlocal flux boundary conditions have been shown to exhibit at most first order convergence to the local counter part asδ → 0, the proposed Neumann-type boundary formulation recovers the local case asO(δ2) in theL∞(Ω) norm, which is optimal considering theO(δ2) convergence of the nonlocal equation to its local limit away from the boundary. We analyze the application of this new boundary treatment to the nonlocal diffusion problem, and present conditions under which the solution of the nonlocal boundary value problem converges to the solution of the corresponding local Neumann problem as the horizon is reduced. To demonstrate the applicability of this nonlocal flux boundary condition to more complicated scenarios, we extend the approach to less regular domains, numerically verifying that we preserve second-order convergence for non-convex domains with corners. Based on the new formulation for nonlocal boundary condition, we develop an asymptotically compatible meshfree discretization, obtaining a solution to the nonlocal diffusion equation with mixed boundary conditions that converges withO(δ2) convergence.

The paper deals with the use of domain decomposition methods in Boundary Domain Integral Method (BDIM) for Navier-Stokes equations of laminar viscous incompressible fluid flow. Since the vorticity transport equation presents the major problem in solving BDIM set of equations, domain decomposition methods are used for its solution. Multiplicative and additive Schwarz iterations are implemented. Partial subdomain problems are computed by the use of Krylov subspace type iterative solvers. Overlapping and non-overlapping subdomain divisions are presented and their effects on convergence of domain decomposition iterative procedures are reported. The new iterative schemes are tested on Poiseuille's flow and Backward facing step flow for various Re number values. 1 Basic theory of Domain Decomposition Method Domain Decomposition Method (DDM) is a relatively old method (Schwarz 1870) and is becoming increasingly popular in modern approximation methods due to high parallelisation capabilities. Similarly as Subdomain technique in BEM it divides the original computational domain into several overlapping or non-overlapping subdomains, which now present partial problems, from which solution over the original domain can be found. The main feature of Domain Decomposition Method is a fact, that solving partial problems is only one step towards the solution of the original system. This follows from a fact, that values of functions and its derivatives on the subdomain interface are not known at the beginning of computation. Through an iterative procedure it is then possible to find the final solution with some combination of calculated values from subdomains. Transactions on Modelling and Simulation vol 9, © 1995 WIT Press, www.witpress.com, ISSN 1743-355X