Cross-Points in Domain Decomposition Methods with a Finite Element Discretization

Non-overlapping domain decomposition methods necessarily have to exchange Dirichlet and Neumann traces at interfaces in order to be able to converge to the underlying mono-domain solution. Well known such non-overlapping methods are the Dirichlet-Neumann method, the FETI and Neumann-Neumann methods, and optimized Schwarz methods. For all these methods, cross-points in the domain decomposition configuration where more than two subdomains meet do not pose any problem at the continuous level, but care must be taken when the methods are discretized. We show in this paper two possible approaches for the consistent discretization of Neumann conditions at cross-points in a Finite Element setting

Non-local optimized Schwarz method with physical boundaries

We extend the theoretical framework of non-local optimized Schwarz methods as introduced in [Claeys,2021], considering an Helmholtz equation posed in a bounded cavity supplemented with a variety of conditions modeling material boundaries. The problem is reformulated equivalently as an equation posed on the skeleton of a non-overlapping partition of the computational domain, involving an operator of the form "identity + contraction". The analysis covers the possibility of resonance phenomena where the Helmholtz problem is not uniquely solvable. In case of unique solvability, the skeleton formulation is proved coercive, and an explicit bound for the coercivity constant is provided in terms of the inf-sup constant of the primary Helmholtz boundary value problem

The optimised Schwarz method and the two-Lagrange multiplier method for heterogeneous problems

In modern science and engineering there exist many heterogeneous problems, in which the material under consideration has non-uniform properties. For example when considering seepage under a dam, water will flow at vastly different rates through sand and stone. Mathematically this can be represented as an elliptic boundary value problem that has a large jump in coefficients between subdomains. The optimised Schwarz method and the related two-Lagrange multiplier method are non-overlapping domain decomposition methods that can be used to numerically solve such boundary value problems. These methods work by solving local Robin problems on each subdomain in parallel, which then piece together to give an approximate solution to the global boundary value problem. It is known that with a careful choice of Robin parameter the convergence of these methods can be sped up. In this thesis we first review the known results for the optimised Schwarz method, deriving optimised Robin parameters and studying the asymptotic performance of the method as the mesh parameter of the discretisation is refined and the jump in coefficients becomes large. Next we formulate the two-Lagrange multiplier method for a model two subdomain problem and show its equivalence to the optimised Schwarz method under suitable conditions. The two-Lagrange multiplier method results in a non-symmetric linear system which is usually solved with a Krylov subspace method such as GMRES. The convergence of the GMRES method can be estimated by constructing a conformal map from the exterior of the field of values of the system matrix to the interior of the unit disc. We approximate the field of values of the two-Lagrange multiplier system matrix by a rectangle and calculate optimised Robin parameters that ensure the rectangle is “well conditioned” in the sense that GMRES converges quickly. We derive convergence estimates for GMRES and consider the behaviour asymptotically as the mesh size is refined and the jump in coefficients becomes large. The final part of the thesis is concerned with the case of heterogeneous problems with many subdomains and cross points, where three or more subdomains coincide. We formulate the two-Lagrange multiplier method for such problems and consider known preconditioners that are needed to improve convergence as the number of subdomains increases. Throughout the thesis numerical experiments are performed to verify the theoretical results.Engineering and Physical Sciences Research Council (EPSRC) grant

A class of alternate strip-based domain decomposition methods for elliptic partial differential Equations

The domain decomposition strategies proposed in this thesis are efficient preconditioning techniques with good parallelism properties for the discrete systems which arise from the finite element approximation of symmetric elliptic boundary value problems in two and three-dimensional Euclidean spaces. For two-dimensional problems, two new domain decomposition preconditioners are introduced, such that the condition number of the preconditioned system is bounded independently of the size of the subdomains and the finite element mesh size. First, the alternate strip-based (ASB2) preconditioner is based on the partitioning of the domain into a finite number of nonoverlapping strips without interior vertices. This preconditioner is obtained from direct solvers inside the strips and a direct fast Poisson solver on the edges between strips, and contains two stages. At each stage the strips change such that the edges between strips at one stage are perpendicular on the edges between strips at the other stage. Next, the alternate strip-based substructuring (ASBS2) preconditioner is a Schur complement solver for the case of a decomposition with multiple nonoverlapping subdomains and interior vertices. The subdomains are assembled into nonoverlapping strips such that the vertices of the strips are on the boundary of the given domain, the edges between strips align with the edges of the subdomains and their union contains all of the interior vertices of the initial decomposition. This preconditioner is produced from direct fast Poisson solvers on the edges between strips and the edges between subdo- mains inside strips, and also contains two stages such that the edges between strips at one stage are perpendicular on the edges between strips at the other stage. The extension to three-dimensional problems is via solvers on slices of the domain

Fundamentally New Coupled Approach to Contact Mechanics via the Dirichlet-Neumann Schwarz Alternating Method

Contact phenomena are essential in understanding the behavior of mechanical systems. Existing computational approaches for simulating mechanical contact often encounter numerical issues, such as inaccurate physical predictions, energy conservation errors, and unwanted oscillations. We introduce an alternative technique, rooted in the non-overlapping Schwarz alternating method, originally developed for domain decomposition. In multi-body contact scenarios, this method treats each body as a separate, non-overlapping domain and prevents interpenetration using an alternating Dirichlet-Neumann iterative process. This approach has a strong theoretical foundation, eliminates the need for contact constraints, and offers flexibility, making it well-suited for multiscale and multi-physics applications. We conducted a numerical comparison between the Schwarz method and traditional methods like Lagrange multiplier and penalty methods, focusing on a benchmark impact problem. Our results indicate that the Schwarz alternating method surpasses traditional methods in several key areas: it provides more accurate predictions for various measurable quantities and demonstrates exceptional energy conservation capabilities. To address the issue of unwanted oscillations in contact velocities and forces, we explored various algorithms and stabilization techniques, ultimately opting for the naive-stabilized Newmark scheme for its simplicity and effectiveness. Furthermore, we validated the efficiency of the Schwarz method in a three-dimensional impact problem, highlighting its innate capacity to accommodate different mesh topologies, time integration schemes, and time steps for each interacting body

A non-overlapping domain decomposition method with high-order transmission conditions and cross-point treatment for Helmholtz problems

International audienceA non-overlapping domain decomposition method (DDM) is proposed for the parallel finite-element solution of large-scale time-harmonic wave problems. It is well-known that the convergence rate of this kind of method strongly depends on the transmission condition enforced on the interfaces between the subdomains. Local conditions based on high-order absorbing boundary conditions (HABCs) have proved to be well-suited, as a good compromise between basic impedance conditions, which lead to suboptimal convergence, and conditions based on the exact Dirichlet-to-Neumann (DtN) map related to the complementary of the subdomain — which are too expensive to compute. However, a direct application of this approach for configurations with interior cross-points (where more than two subdomains meet) and boundary cross-points (points that belong to both the exterior boundary and at least two subdomains) is suboptimal and, in some cases, can lead to incorrect results.In this work, we extend a non-overlapping DDM with HABC-based transmission conditions approach to efficiently deal with cross-points for lattice-type partitioning. We address the question of the cross-point treatment when the HABC operator is used in the transmission condition, or when it is used in the exterior boundary condition, or both. The proposed cross-point treatment relies on corner conditions developed for Padé-type HABCs. Two-dimensional numerical results with a nodal finite-element discretization are proposed to validate the approach, including convergence studies with respect to the frequency, the mesh size and the number of subdomains. These results demonstrate the efficiency of the cross-point treatment for settings with regular partitions and homogeneous media. Numerical experiments with distorted partitions and smoothly varying heterogeneous media show the robustness of this treatment

ICASE

This report summarizes research conducted at the Institute for Computer Applications in Science and Engineering in the areas of (1) applied and numerical mathematics, including numerical analysis and algorithm development; (2) theoretical and computational research in fluid mechanics in selected areas of interest, including acoustics and combustion; (3) experimental research in transition and turbulence and aerodynamics involving Langley facilities and scientists; and (4) computer science