Two-level Additive Schwarz Preconditioner Research Articles

Robust domain decomposition methods for high-contrast multiscale problems on irregular domains with virtual element discretizations

Our research focuses on the development of domain decomposition preconditioners tailored for second-order elliptic partial differential equations. Our approach addresses two major challenges simultaneously: i) effectively handling coefficients with high-contrast and multiscale properties, and ii) accommodating irregular domains in the original problem, the coarse mesh, and the subdomain partition. The robustness of our preconditioners is crucial for real-world applications, such as the efficient and accurate modeling of subsurface flow in porous media and other important domains.The core of our method lies in the construction of a suitable partition of unity functions and coarse spaces utilizing local spectral information. Leveraging these components, we implement a two-level additive Schwarz preconditioner. We demonstrate that the condition number of the preconditioned systems is bounded with a bound that is independent of the contrast. Our claims are further substantiated through selected numerical experiments, which confirm the robustness of our preconditioners.

A two-level additive Schwarz preconditioner for the Nitsche extended finite element approximation of elliptic interface problems

A two-level additive Schwarz preconditioner for the Nitsche extended finite element approximation of elliptic interface problems

Two-level additive Schwarz methods for three-dimensional unsteady Stokes flows in patient-specific arteries with parameterized one-dimensional central-line coarse preconditioner

We consider the numerical solution of unsteady Stokes equations in patient-specific arterial-like domains in 3D. A Stokes-like solver is a necessary component in a more sophisticated nonlinear Navier-Stokes method, for which several multilevel domain decomposition methods have been introduced recently. Because of the complex geometry, the construction and the solve of the coarse problem usually take a large percentage of the total compute time. In this paper, we introduce a parameterized one-dimensional Stokes solver defined along the centerline of the artery and use its stabilized finite element discretization to construct a coarse preconditioner. With suitable 3D-to-1D restriction and 1D-to-3D extension operators on fully unstructured meshes, a two-level additive Schwarz preconditioner can be constructed. Some numerical experiments for flows in realistic arteries are presented to show the efficiency and robustness of the new coarse preconditioner whose computational cost is considerably lower than the existing three-dimensional coarse preconditioners.

Preconditioning for finite element methods with strain smoothing

Strain smoothing methods such as the smoothed finite element methods (S-FEMs) and the strain-smoothed element~(SSE) method have successfully improved the performance of finite elements, and there have been numerous applications of them in finite element analysis. For the sake of efficient applications to large-scale problems, it is important to develop a mathematically and numerically well-elaborated iterative solver for the strain smoothing methods. In this paper, inspired by the spectral properties of the strain smoothing methods, we propose efficient ways of preconditioning for the methods. First, we analyze the spectrums of the stiffness matrices of the edge-based S-FEM and the SSE method. Then, we propose an improved two-level additive Schwarz preconditioner for the strain smoothing methods by modifying local solvers appropriately. For the sake of convenience of implementation, an alternative form of the preconditioner is also proposed by defining the coarse-scale operation in terms of the standard FEM. We verify our theoretical results through numerical experiments.

A central-line coarse preconditioner for Stokes flows in artery-like domains

We consider numerical simulation of blood flows in the artery using multilevel domain decomposition methods. Because of the complex geometry, the construction and the solve of the coarse problem take a large percentage of the total compute time in the multilevel method. In this paper, we introduce a one-dimensional central-line model of the blood flow and use its stabilized finite element discretization to construct a coarse preconditioner. With suitable restriction and extension operators, we obtain a two-level additive Schwarz preconditioner for two- and three-dimensional problems. We present some numerical experiments with different arteries to show the efficiency and robustness of the new coarse preconditioner whose computational cost is considerably lower than other coarse preconditioners constructed using the two- or three-dimensional geometry of the artery.

High-performance dune modules for solving large-scale, strongly anisotropic elliptic problems with applications to aerospace composites

The key innovation in this paper is an open-source, high-performance iterative solver for high contrast, strongly anisotropic elliptic partial differential equations implemented within dune-pdelab. The iterative solver exploits a robust, scalable two-level additive Schwarz preconditioner, GenEO (Spillane et al., 2014). The development of this solver has been motivated by the need to overcome the limitations of commercially available modelling tools for solving structural analysis simulations in aerospace composite applications. Our software toolbox dune-composites encapsulates the mathematical complexities of the underlying packages within an efficient C++ framework, providing an application interface to our new high-performance solver. We illustrate its use on a range of industrially motivated examples, which should enable other scientists to build on and extend dune-composites and the GenEO preconditioner for use in their own applications. We demonstrate the scalability of the solver on more than 15,000 cores of the UK national supercomputer Archer, solving an aerospace composite problem with over 200 million degrees of freedom in a few minutes. This scale of computation brings composites problems that would otherwise be unthinkable into the feasible range. To demonstrate the wider applicability of the new solver, we also confirm the robustness and scalability of the solver on SPE10, a challenging benchmark in subsurface flow/reservoir simulation. Program summaryProgram Title:dune-compositesProgram Files doi:http://dx.doi.org/10.17632/96mtdcmjsb.1Licensing provisions: BSD 3-clauseProgramming language: C++Nature of problem:dune-composites is designed to solve anisotropic linear elasticity equations for anisotropic, heterogeneous materials, e.g. composite materials. To achieve this, our contribution also implements a new preconditioner in dune-pdelab.Solution method: The anisotropic elliptic partial differential equations are solved via the finite element method. The resulting linear system is solved via an iterative solver with a robust, scalable two-level overlapping Schwarz preconditioner: GenEO.

A Two-Level Additive Schwarz Preconditioner for Local $$C^0$$ C 0 Discontinuous Galerkin Methods of Kirchhoff Plates

A two-level additive Schwarz preconditioner based on the overlapping domain decomposition approach is proposed for the local $$C^0$$ discontinuous Galerkin (LCDG) method of Kirchhoff plates. Then with the help of an intergrid transfer operator and its error estimates, it is proved that the condition number is bounded by $$O(1+(H^4/\delta ^4))$$ , where H is the diameter of the subdomains and $$\delta $$ measures the overlap among subdomains. And for some special cases of small overlap, the estimate can be improved as $$O(1+(H^3/\delta ^3))$$ . At last, some numerical results are reported to demonstrate the high efficiency of the two-level additive Schwarz preconditioner.

English

The finite element (FE) solution of geotechnical elasticity problems leads to the solution of a large system of linear equations. For solving the system, we use the preconditioned conjugate gradient (PCG) method with two-level additive Schwarz preconditioner. The preconditioning is realised in parallel. A coarse space is usually constructed using an aggregation technique. If the finite element spaces for coarse and fine problems on structural grids are fully compatible, relations between elements of matrices of the coarse and fine problems can be derived. By generalization of these formulae, we obtain an overlapping aggregation technique for the construction of a coarse space with smoothed basis functions. The numerical tests are presented at the end of the paper.

Domain decomposition methods for space fractional partial differential equations

A two-level additive Schwarz preconditioner is proposed for solving the algebraic systems resulting from the finite element approximations of space fractional partial differential equations (SFPDEs). It is shown that the condition number of the preconditioned system is bounded by C(1+H/δ), where H is the maximum diameter of subdomains and δ is the overlap size among the subdomains. Numerical results are given to support our theoretical findings.

Domain decomposition preconditioners for the discontinuous Petrov–Galerkin method

In this paper, we design some efficient domain decomposition preconditioners for the discontinuous Petrov–Galerkin (DPG) method. Due to the special properties of the DPG method, the boundary condition becomes crucial in both of its application and analysis. We mainly focus on one of the boundary conditions: the Robin boundary condition, which actually appears in some useful model problems like the Helmholtz equation. We first design a two-level additive Schwarz preconditioner for the Poisson equation with a Robin boundary condition and give a rigorous condition number estimate for the preconditioned algebraic system. Moreover we also construct an additive Schwarz preconditioner for solving the Helmholtz equation. Numerical results show that the condition number of the preconditioned system is independent of wavenumber ω and mesh size h .

Schwarz preconditioners for stochastic elliptic PDEs

Increasingly the spectral stochastic finite element method (SSFEM) has become a popular computational tool for uncertainty quantification in numerous practical engineering problems. For large-scale problems however, the computational cost associated with solving the arising linear system in the SSFEM still poses a significant challenge. The development of efficient and robust preconditioned iterative solvers for the SSFEM linear system is thus of paramount importance for uncertainty quantification of large-scale industrially relevant problems. In the context of high performance computing, the preconditioner must scale to a large number of processors. Therefore in this paper, a two-level additive Schwarz preconditioner is described for the iterative solution of the SSFEM linear system. The proposed preconditioner can be viewed as a generalization of the mean based block-diagonal preconditioner commonly used in the literature. For the numerical illustrations, two-dimensional steady-state diffusion and elasticity problems with spatially varying random coefficients are considered. The performance of the algorithm is investigated with respect to the geometric parameters, strength of randomness, dimension and order of the stochastic expansion.

Two-Level Additive Schwarz Preconditioners for a Weakly Over-Penalized Symmetric Interior Penalty Method

We propose and analyze several two-level additive Schwarz preconditioners for a weakly over-penalized symmetric interior penalty method for second order elliptic boundary value problems. We also report numerical results that illustrate the parallel performance of these preconditioners.

Additive Schwarz with aggregation-based coarsening for elliptic problems with highly variable coefficients

We develop a new coefficient-explicit theory for two-level overlapping domain decomposition preconditioners with non-standard coarse spaces in iterative solvers for finite element discretisations of second-order elliptic problems. We apply the theory to the case of smoothed aggregation coarse spaces introduced by Vanek, Mandel and Brezina in the context of algebraic multigrid (AMG) and are particularly interested in the situation where the diffusion coefficient (or the permeability) α is highly variable throughout the domain. Our motivating example is Monte Carlo simulation for flow in rock with permeability modelled by log–normal random fields. By using the concept of strong connections (suitably adapted from the AMG context) we design a two-level additive Schwarz preconditioner that is robust to strong variations in α as well as to mesh refinement. We give upper bounds on the condition number of the preconditioned system which do not depend on the size of the subdomains and make explicit the interplay between the coefficient function and the coarse space basis functions. In particular, we are able to show that the condition number can be bounded independent of the ratio of the two values of α in a binary medium even when the discontinuities in the coefficient function are not resolved by the coarse mesh. Our numerical results show that the bounds with respect to the mesh parameters are sharp and that the method is indeed robust to strong variations in α. We compare the method to other preconditioners and to a sparse direct solver.

Optimal left and right additive Schwarz preconditioning for minimal residual methods with Euclidean and energy norms

For the solution of non-symmetric or indefinite linear systems arising from discretizations of elliptic problems, two-level additive Schwarz preconditioners are known to be optimal in the sense that convergence bounds for the preconditioned problem are independent of the mesh and the number of subdomains. These bounds are based on some kind of energy norm. However, in practice, iterative methods which minimize the Euclidean norm of the residual are used, despite the fact that the usual bounds are non-optimal, i.e., the quantities appearing in the bounds may depend on the mesh size; see [X.-C. Cai, J. Zou, Some observations on the l 2 convergence of the additive Schwarz preconditioned GMRES method, Numer. Linear Algebra Appl. 9 (2002) 379–397]. In this paper, iterative methods are presented which minimize the same energy norm in which the optimal Schwarz bounds are derived, thus maintaining the Schwarz optimality. As a consequence, bounds for the Euclidean norm minimization are also derived, thus providing a theoretical justification for the practical use of Euclidean norm minimization methods preconditioned with additive Schwarz. Both left and right preconditioners are considered, and relations between them are derived. Numerical experiments illustrate the theoretical developments.

Two-level additive Schwarz preconditioners for C0 interior penalty methods

We study two-level additive Schwarz preconditioners that can be used in the iterative solution of the discrete problems resulting from C0 interior penalty methods for fourth order elliptic boundary value problems. We show that the condition number of the preconditioned system is bounded by C(1+(H3/δ3)), where H is the typical diameter of a subdomain, δ measures the overlap among the subdomains and the positive constant C is independent of the mesh sizes and the number of subdomains.

An Improved Convergence Bound for Aggregation-Based Domain Decomposition Preconditioners

In this paper we present a two-level overlapping domain decomposition preconditioner for the finite-element discretization of elliptic problems in two and three dimensions. The computational domain is partitioned into overlapping subdomains, and a coarse space correction, based on aggregation techniques, is added. Our definition of the coarse space does not require the introduction of a coarse grid. We consider a set of assumptions on the coarse basis functions to bound the condition number of the resulting preconditioned system. These assumptions involve only geometrical quantities associated with the aggregates and the subdomains. We prove that the condition number using the two-level additive Schwarz preconditioner is $\mathcal{O}(H/\delta + H_0/\delta)$, where H and $H_0$ are the diameters of the subdomains and the aggregates, respectively, and $\delta$ is the overlap among the subdomains and the aggregates. This extends the bounds presented in [C. Lasser and A. Toselli, Convergence of some two-level overlapping domain decomposition preconditioners with smoothed aggregation coarse spaces, in Recent Developments in Domain Decomposition Methods, Lecture Notes in Comput. Sci. Engrg. 23, L. Pavarino and A. Toselli, eds., Springer-Verlag, Berlin, 2002, pp. 95--117; M. Sala, Domain Decomposition Preconditioners: Theoretical Properties, Application to the Compressible Euler Equations, Parallel Aspects, Ph.D. thesis, Ecole Polytechnique Fédérale de Lausanne, Lausanne, Switzerland, 2003; M. Sala, Math. Model. Numer. Anal., 38 (2004), pp. 765--780]. Numerical experiments on a model problem are reported to illustrate the performance of the proposed preconditioner.

Two-Level Additive Schwarz Preconditioners for the h - p Version of the Galerkin Boundary Element Method for 2-d Problems

We study two-level additive Schwarz preconditioners for the h-p version of the Galerkin boundary element method when used to solve hypersingular integral equations of the first kind, which arise from the Neumann problems for the Laplacian in two dimensions. Overlapping and non-overlapping methods are considered. We prove that the non-overlapping preconditioner yields a system of equations having a condition number bounded by c(1 + log p)2 maxi + logHi/hi) where Hi is the length of the i-th subdomain, hi is the maximum length of the elements in this subdomain, and p is the maximum polynomial degree used. For the overlapping method, we prove that the condition number is bounded by c(l + log H/z (I + logp)2 where z is the size of the overlap and H = maxi Hi. We also discuss the use of the non-overlapping method when the mesh is geometrically graded. The condition number in that case is bounded by c log2 M, where M is the degrees of freedom.

Lower Bounds for Two-Level Additive Schwarz Preconditioners with Small Overlap

Lower bounds for the condition numbers of the preconditioned systems are obtained for two-level additive Schwarz preconditioners. They show that the known upper bounds for both second order and fourth order problems are sharp in the case of a small overlap.

Two-level additive Schwarz preconditioners for P1 nonconforming finite elements for nonsymmetric and indefinite problems

Two-level additive Schwarz preconditioners are developed for P1 nonconforming finite element approximation of second-order nonsymmetric and indefinite elliptic problems. The convergence factor is shown to be independent of the number of subspaces when this preconditioner is used with GMRES method.

Two-level additive Schwarz preconditioners for plate elements

We present a theory of two-level additive Schwarz preconditioners for plate elements. It is applicable to the cases where the finite element spaces on the two levels are (I) conforming and nested, (II) conforming and nonnested, and (III) nonconforming and nonnested. The condition number of the preconditioned system is independent of the mesh sizes and the number of subdomains in the case of generous overlap.