Subdomain Problems Research Articles

Optimized Schwarz waveform relaxation method for the incompressible Stokes problem

We propose and analyse the optimized Schwarz waveform relaxation (OSWR) method for the unsteady incompressible Stokes equations. Well-posedness of the local subdomain problems with Robin boundary conditions is proved. Convergence of the velocity is shown through energy estimates; however, pressure converges only up to constant values in the subdomains, and an astute correction technique is proposed to recover these constants from the velocity. The convergence factor of the OSWR algorithm is obtained through a Fourier analysis, and allows to efficiently optimize the space-time Robin transmission conditions involved in the OSWR method. Then, numerical illustrations for the two-dimensional unsteady incompressible Stokes system are presented to illustrate the performance of the OSWR algorithm.

An optimisation–based domain–decomposition reduced order model for the incompressible Navier-Stokes equations

The aim of this work is to present a model reduction technique in the framework of optimal control problems for partial differential equations. We combine two approaches used for reducing the computational cost of the mathematical numerical models: domain–decomposition (DD) methods and reduced–order modelling (ROM). In particular, we consider an optimisation–based domain–decomposition algorithm for the parameter–dependent stationary incompressible Navier–Stokes equations. Firstly, the problem is described on the subdomains coupled at the interface and solved through an optimal control problem, which leads to the complete separation of the subdomain problems in the DD method. On top of that, a reduced model for the obtained optimal–control problem is built; the procedure is based on the Proper Orthogonal Decomposition technique and a further Galerkin projection. The presented methodology is tested on two fluid dynamics benchmarks: the stationary backward–facing step and lid-driven cavity flow. The numerical tests show a significant reduction of the computational costs in terms of both the problem dimensions and the number of optimisation iterations in the domain–decomposition algorithm.

On the Bahri–Lions conjecture for elliptic equations with non-symmetric nonlinearities

We prove the existence of infinitely many solutions for a class of elliptic Dirichlet problems with non-symmetric nonlinearities. In particular, in suitable domains of {mathbb {R}}^n with nge 3, this result gives a positive answer to a well known conjecture formulated by A. Bahri and P.L. Lions. The proof is based on a minimization method which does not require the use of techniques of deformation from the symmetry. This method allows us to piece together solutions of Dirichlet problems in suitable subdomains, so we obtain infinitely many nodal solutions with a prescribed nodal structure.

A preconditioning method with auxiliary crack tip subproblems for dynamic crack propagation based on XFEM

In this paper, a domain decomposition based preconditioning method is developed to accelerate the Krylov subspace method for solving the linear system arising from the extended finite element discretization of the dynamic crack problem. Based on the observation that the crack tip area has a significant impact on the convergence of the iterative method while the other mesh points do not, the finite element mesh is partitioned into two types: the regular subdomains and the crack tip subdomains. To construct the additive Schwarz preconditioner, the global matrix is partitioned into submatrices which are solved exactly in the crack tip subdomains by the LU factorization and inexactly in the regular subdomains by an incomplete LU factorization. As the crack propagates, we develop a scheme to update the subdomain problems without resolving of them, in which, in order to save the computational cost, only the subdomains around the crack are updated, and all the other regular subdomains remain unchanged. To further speed up the Krylov subspace method, an auxiliary subproblem including crack tips of the previous and current time steps is constructed and solved to provide a better initial guess. We carefully studied the properties of the linear system and the performance of the proposed algorithm. The numerical experiments indicate that the proposed method works well for the simulation of dynamic crack propagation with multiple crack tips.

A Space-Time Multiscale Mortar Mixed Finite Element Method for Parabolic Equations

We develop a space-time mortar mixed finite element method for parabolic problems. The domain is decomposed into a union of subdomains discretized with nonmatching spatial grids and asynchronous time steps. The method is based on a space-time variational formulation that couples mixed finite elements in space with discontinuous Galerkin in time. Continuity of flux (mass conservation) across space-time interfaces is imposed via a coarse-scale space-time mortar variable that approximates the primary variable. Uniqueness, existence, and stability as well as a priori error estimates for the spatial and temporal errors are established. A space-time nonoverlapping domain decomposition method is developed that reduces the global problem to a space-time coarse-scale mortar interface problem. Each interface iteration involves solving in parallel space-time subdomain problems. The spectral properties of the interface operator and the convergence of the interface iteration are analyzed. Numerical experiments are provided that illustrate the theoretical results and the flexibility of the method for modeling problems with features that are localized in space and time.

Interface flux recovery framework for constructing partitioned heterogeneous time‐integration methods

AbstractA common approach for the development of partitioned schemes employing different time integrators on different subdomains is to lag the coupling terms in time. This can lead to accuracy issues, especially in multistage methods. In this article, we present a novel framework for partitioned heterogeneous time‐integration methods, which allows the coupling of arbitrary multistage and multistep methods without reducing their order of accuracy. At the core of our approach are accurate estimates of the interface flux obtained from the Schur complement of an auxiliary monolithic system. We use these estimates to construct a polynomial‐in‐time approximation of the interface flux over the current time coupling window. This approximation provides the interface boundary conditions necessary to decouple the subdomain problems at any point within the coupling window. In so doing our framework enables a flexible choice of time‐integrators for the individual subproblems without compromising the time‐accuracy at the coupled problem level. This feature is the main distinction between our framework and other approaches. To demonstrate the framework, we construct a family of partitioned heterogeneous time‐integration methods, combining multistage and multistep methods, for a simplified tracer transport component of the coupled air‐sea system in Earth system models. We report numerical tests evaluating accuracy and flux conservation for different pairs of time‐integrators from the explicit Runge‐Kutta and Adams‐Moulton families.

Flux-mortar mixed finite element methods with multipoint flux approximation

The flux-mortar mixed finite element method was recently developed in Boon et al. (2022) for a general class of domain decomposition saddle point problems on non-matching grids. In this work we develop the method for Darcy flow using the multipoint flux approximation as the subdomain discretization. The subdomain problems involve solving positive definite cell-centered pressure systems. The normal flux on the subdomain interfaces is the mortar coupling variable, which plays the role of a Lagrange multiplier to impose weakly continuity of pressure. We present well-posedness and error analysis based on reformulating the method as a mixed finite element method with a quadrature rule. We develop a non-overlapping domain decomposition algorithm for the solution of the resulting algebraic system that reduces it to an interface problem for the flux-mortar, as well as an efficient interface preconditioner. A series of numerical experiments is presented illustrating the performance of the method on general grids, including applications to flow in complex porous media.

BDDC Algorithms for Oseen problems with HDG Discretizations

Abstract The balancing domain decomposition by constraints (BDDC) methods are applied to the linear system arising from the hybridizable discontinuous Galerkin (HDG) discretization of the Oseen equation of incompressible flow. The generalized minimal residual method (GMRES) is used to accelerate the convergence. The original system is first reduced to a subdomain interface problem that is asymmetric indefinite, but can be positive definite in a special subspace. Edge/face average constraints can ensure all BDDC-preconditioned GMRES iterates stay in this special subspace. The convergence of the algorithm is analyzed, and additional edge/face constraints are used to improve the convergence. If the subdomain size is small enough, the number of iterations is independent of the number of subdomains, and depends only slightly on the subdomain problem size when the viscosity is large. When the viscosity is small, the convergence deteriorates. However, the numerical examples can give satisfactory results for the linear HDG discretization even with a small viscosity.

A convolutional plane wave model for sound field reconstruction

Spatial sound field interpolation relies on suitable models to conform to available measurements and predict the sound field in the domain of interest. A suitable model can be difficult to determine when the spatial domain of interest is large compared to the wavelength or when spherical and planar wavefronts are present or the sound field is complex, as in the near-field. To span such complex sound fields, the global reconstruction task can be partitioned into local subdomain problems. Previous studies have shown that partitioning approaches rely on sufficient measurements within each domain due to the higher number of model coefficients. This study proposes a joint analysis of all of the local subdomains while enforcing self-similarity between neighbouring partitions. More specifically, the coefficients of local plane wave representations are sought to have spatially smooth magnitudes. A convolutional model of the sound field in terms of plane wave filters is formulated and the inverse reconstruction problem is solved via the alternating direction method of multipliers. The experiments on simulated and measured sound fields suggest that the proposed method retains the flexibility of local models to conform to complex sound fields and also preserves the global structure to reconstruct from fewer measurements.

Fully implicit local time-stepping methods for advection-diffusion problems in mixed formulations

This paper is concerned with numerical solution of transport problems in heterogeneous porous media. A semi-discrete continuous-in-time formulation of the linear advection-diffusion equation is obtained by using a mixed hybrid finite element method, in which the flux variable represents both the advective and diffusive flux, and the Lagrange multiplier arising from the hybridization is used for the discretization of the advective term. Based on global-in-time and nonoverlapping domain decomposition, we propose two implicit local time-stepping methods to solve the semi-discrete problem. The first method uses the time-dependent Steklov-Poincaré type operator and the second uses the optimized Schwarz waveform relaxation (OSWR) with Robin transmission conditions. For each method, we formulate a space-time interface problem which is solved iteratively. Each iteration involves solving the subdomain problems independently and globally in time; thus, different time steps can be used in the subdomains. The convergence of the fully discrete OSWR algorithm with nonmatching time grids is proved. Numerical results for problems with various Peclét numbers and discontinuous coefficients, including a prototype for the simulation of the underground storage of nuclear waste, are presented to illustrate the performance of the proposed local time-stepping methods.

A parallel scalable domain decomposition preconditioner for elastic crack simulation using XFEM

AbstractIn this article, a parallel overlapping domain decomposition preconditioner is proposed to solve the linear system of equations arising from the extended finite element discretization of elastic crack problems. The algorithm partitions the computational mesh into two types of subdomains: the regular subdomains and the crack tip subdomains based on the observation that the crack tips have a significant impact on the convergence of the iterative method while the impact of the crack lines is not that different from those of regular mesh points. The tip subdomains consist of mesh points at crack tips and all neighboring points where the branch enrichment functions are applied. The regular subdomains consist of all other mesh points, including those on the crack lines. To overcome the mismatch between the number of subdomains and the number of processor cores, the proposed method is divided into two steps: solve the crack tip problem and then the regular subdomain problem during each iteration. The proposed method was used to develop a parallel XFEM package which is able to test different types of iterative methods. To achieve good parallel efficiency, additional methods were introduced to reduce communication and to maintain the load balance between processors. Numerical experiments indicate that the proposed method significantly reduces the number of iterations and the total computation time compared to the classical methods. In addition, the method scales up to 8192 processor cores with over 70% parallel efficiency to solve problems with more than degrees of freedom.

Domain decomposition and expanded mixed method for parabolic partial differential equations

We developed and analyzed the multiblock mortar expanded mixed method for second order parabolic partial differential equations. This is a domain decomposition method in which the computational domain is expressed as the union of non-overlapping subdomains separated by interfaces. An auxiliary variable is introduced on the interface which represents the pressure and serves as Dirichlet boundary condition for local subdomain problems. The interface variable also plays the part of Lagrange multiplier to enforce flux matching condition on the interfaces. We explored the expanded mixed method to discretize each subdomain. We propose the semi-discrete formulation and address the solvability of the discrete problem. The optimal order convergence is provided for the continuous time case. We also investigate the fully discrete formulation and derived corresponding error estimates. The numerical experiments are conducted to demonstrate the theory developed in the paper.

Optimized Ventcel-Schwarz waveform relaxation and mixed hybrid finite element method for transport problems

<p style='text-indent:20px;'>This paper is concerned with the optimized Schwarz waveform relaxation method and Ventcel transmission conditions for the linear advection-diffusion equation. A mixed formulation is considered in which the flux variable represents both diffusive and advective flux, and Lagrange multipliers are introduced on the interfaces between nonoverlapping subdomains to handle tangential derivatives in the Ventcel conditions. A space-time interface problem is formulated and is solved iteratively. Each iteration involves the solution of time-dependent problems with Ventcel boundary conditions in the subdomains. The subdomain problems are discretized in space by a mixed hybrid finite element method based on the lowest-order Raviart-Thomas space and in time by the backward Euler method. The proposed algorithm is fully implicit and enables different time steps in the subdomains. Numerical results with discontinuous coefficients and various Peclét numbers validate the accuracy of the method with nonconforming time grids and confirm the improved convergence properties of Ventcel conditions over Robin conditions.</p>

A recycling preconditioning method with auxiliary tip subspace for elastic crack propagation simulation using XFEM

In this paper, we propose a recycling preconditioning method with auxiliary tip subspace for solving a sequence of highly ill-conditioned linear systems of equations of different sizes arising from elastic crack propagation problems discretized by the extended finite element method. To construct a Schwarz type preconditioner, the finite element mesh is decomposed into crack tip subdomains, which contain all the degrees of freedom (DOFs) of the branch enrichment functions, and regular subdomains, which contain the standard DOFs and the DOFs of the Heaviside and the Junction enrichment functions. As cracks propagate these subdomains are modified accordingly, and the subdomain matrices are constructed as the restriction of the global matrix to the subdomains. In the overlapping Schwarz preconditioners, the crack tip subproblems are solved exactly and the regular subproblems are solved by some inexact solvers, such as ILU. We consider problems with and without crack intersections and develop a simple scheme to update, instead of re-computing, the subdomain problems as cracks propagate, in which only crack tip subdomains are updated around the new crack tips and all the regular subdomains remain unchanged. Therefore, no extra search is required, and the sizes of crack tip subproblems do not increase as cracks propagate, which greatly saves the computational cost. Moreover, starting from the second system, the Krylov subspace method uses a nontrivial initial guess constructed using the solution of the previous system with a modification around the new crack tips. The strategy accelerates the convergence remarkably. Numerical experiments demonstrate the efficiency of the proposed algorithms applied to problems with several types of cracks.

BDDC algorithms for advection-diffusion problems with HDG discretizations

In this paper, a preconditioned GMRES method is developed and analyzed for solving the linear system from advection-diffusion equations with the hybridizable discontinuous Galerkin (HDG) discretization. The preconditioner is the balancing domain decomposition methods (BDDC), one of the most popular nonoverlapping domain decomposition methods. For large viscosity, if the subdomain size is small enough, the number of iterations is independent of the number of subdomains and depends only slightly on the subdomain problem size. The convergence deteriorates when the viscosity decreases. These results are similar to those with the standard finite element discretizations. Numerical results of two examples in two dimensions are provided to confirm the theory.

Combining machine learning and domain decomposition methods for the solution of partial differential equations—A review

AbstractScientific machine learning (SciML), an area of research where techniques from machine learning and scientific computing are combined, has become of increasing importance and receives growing attention. Here, our focus is on a very specific area within SciML given by the combination of domain decomposition methods (DDMs) with machine learning techniques for the solution of partial differential equations. The aim of the present work is to make an attempt of providing a review of existing and also new approaches within this field as well as to present some known results in a unified framework; no claim of completeness is made. As a concrete example of machine learning enhanced DDMs, an approach is presented which uses neural networks to reduce the computational effort in adaptive DDMs while retaining their robustness. More precisely, deep neural networks are used to predict the geometric location of constraints which are needed to define a robust coarse space. Additionally, two recently published deep domain decomposition approaches are presented in a unified framework. Both approaches use physics‐constrained neural networks to replace the discretization and solution of the subdomain problems of a given decomposition of the computational domain. Finally, a brief overview is given of several further approaches which combine machine learning with ideas from DDMs to either increase the performance of already existing algorithms or to create completely new methods.

Nonoverlapping domain decomposition method with preconditioner from asymptotic analysis of steady flow in high contrast media

ABSTRACT We present a nonoverlapping domain decomposition method for steady flow in high contrast heterogeneous media modelled by an elliptic equation with coefficients that have very large amplitude variations on a small spatial scale. The linear system of equations resulting from matching the solution trace and the fluxes through the boundary of the subdomains is ill-conditioned, especially for fine meshes needed to capture the rapid variations of the solution. Our main contribution is to show with analysis and numerical simulations how to use an asymptotic approximation of the Dirichlet to Neumann map of the sub-domain problems to obtain a preconditioner for an efficient domain decomposition algorithm.

Optimized Schwarz Methods with Elliptical Domain Decompositions

Over the past decade, partial differential equation models in elliptical geometries have become a focus of interest in several scientific and engineering applications: the classical studies of flow past a cylinder, the spherical particles in nano-fluids and spherical water filled domains are replaced by elliptical geometries which more accurately describe a wider class of physical problems of interest. Optimized Schwarz methods (OSMs) are among the best parallel methods for such models. We study here for the first time OSMs with elliptical domain decompositions, i.e. decompositions into an ellipse and elliptical rings. Using the technique of separation of variables, we decouple the spatial variables and reduce the subdomain problems to radial Mathieu like equations defined on finite intervals, which allows us to derive and study a new family of OSMs. Our analysis reveals that the optimized transmission parameters are not constants any more along the elliptical interfaces. We can prove however also that using the constant optimized parameters from the straight interface analysis in the literature scaled locally by the interface curvature is still efficient in an asymptotic sense, which leads to the important discovery of a unique factor in the optimized parameters and asymptotic performance determined by the geometry of the decomposition. We use numerical examples to illustrate our analysis and findings.

Smoothness of solutions to the mixed problem for elliptic differential-difference equation in cylinder

This article deals with mixed boundary value problem for an elliptic differential-difference equation in a cylinder. There are obtained results on the smoothness of generalized solutions of such problem in subdomains, and necessary and sufficient conditions for the preservation of smoothness on the boundaries of neighboring subdomains.

Space–time domain decomposition for two-phase flow between different rock types

In Ahmed et al. (2019) a space–time domain decomposition method was proposed for two-phase flow in a porous medium composed of two different rock types, so that the capillary pressure field is discontinuous at the interface between the rocks. For this nonlinear and degenerate parabolic problem, with nonlinear and discontinuous transmission conditions on the interface, the Optimized Schwarz waveform relaxation method (OSWR) with Robin or Ventcell transmission conditions was considered. A guaranteed and fully computable a posteriori error estimate was derived, which in particular took into account the domain decomposition error.In this paper we provide a mathematical and numerical analysis of this space–time domain decomposition method in the Robin case. Complete numerical approximation is achieved by a finite volume scheme in space and the lowest order discontinuous Galerkin method in time. We prove the existence of a weak solution of the two-phase flow subdomain problem with Robin boundary conditions by analyzing the convergence of the finite volume scheme. The domain decomposition algorithm is based on the solution of space–time nonlinear subdomain problems over the whole time interval, allowing for different time steps in different parts of the domain, adapted to the physical properties of each subdomain, and we show that such an algorithm is well-defined. Numerical experiments on three-dimensional problems with different rock types illustrate the performance of the domain decomposition method.