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A new reduced-dimension adaptive generalized Dryja-Smith-Widlund (GDSW) overlapping Schwarz method for linear second-order elliptic problems in three dimensions is introduced. It is robust with respect to large contrasts of the coefficients of the partial differential equations. The condition number bound of the new method is shown to be independent of the coefficient contrast and only dependent on a user-prescribed tolerance. The interface of the nonoverlapping domain decomposition is partitioned into nonoverlapping patches. The new coarse space is obtained by selecting a few eigenvectors of certain local eigenproblems which are defined on these patches. These eigenmodes are energy-minimally extended to the interior of the nonoverlapping subdomains and added to the coarse space. By using a new interface decomposition, the reduced-dimension adaptive GDSW overlapping Schwarz method usually has a smaller coarse space than existing GDSW and adaptive GDSW domain decomposition methods. A robust condition number estimate is proven for the new reduced-dimension adaptive GDSW method which is also valid for existing adaptive GDSW methods. Numerical results for the equations of isotropic linear elasticity in three dimensions confirming the theoretical findings are presented.

A robust two-level overlapping Schwarz method for scalar elliptic model problems with highly varying coefficient functions is introduced. While the convergence of standard coarse spaces may depend strongly on the contrast of the coefficient function, the condition number bound of the new method is independent of the coefficient function. Indeed, the condition number only depends on a user-prescribed tolerance. The coarse space is based on discrete harmonic extensions of vertex, edge, and face interface functions, which are computed from the solutions of corresponding local generalized edge and face eigenvalue problems. The local eigenvalue problems are of the size of the edges and faces of the decomposition, and the eigenvalue problems can be constructed solely from the local subdomain stiffness matrices and the fully assembled global stiffness matrix. The new AGDSW (adaptive generalized Dryja-Smith-Widlund) coarse space always contains the classical GDSW coarse space by construction of the generalized eigenvalue problems. Numerical results supporting the theory are presented for several model problems in three dimensions using structured as well as unstructured meshes and unstructured decompositions.

A robust two-level overlapping Schwarz method for scalar elliptic model problems with highly varying coefficient functions is introduced. While the convergence of standard coarse spaces may depend strongly on the contrast of the coefficient function, the condition number bound of the new method is independent of the coefficient function. Indeed, the condition number only depends on a user-prescribed tolerance. The coarse space is based on discrete harmonic extensions of vertex, edge, and face interface functions, which are computed from the solutions of corresponding local generalized edge and face eigenvalue problems. The local eigenvalue problems are of the size of the edges and faces of the decomposition, and the eigenvalue problems can be constructed solely from the local subdomain stiffness matrices and the fully assembled global stiffness matrix. The new AGDSW (adaptive generalized Dryja-Smith-Widlund) coarse space always contains the classical GDSW coarse space by construction of the generalized eigenvalue problems. Numerical results supporting the theory are presented for several model problems in three dimensions using structured as well as unstructured meshes and unstructured decompositions.

Parallel Newton-Krylov FETI-DP (Finite Element Tearing and Interconnecting-Dual-Primal) domain decomposition methods are fast and robust solvers, e.g., for nonlinear implicit problems in structural mechanics. In these methods, the nonlinear problem is first linearized and then decomposed into loosely coupled (linear) problems, which can be solved in parallel. By changing the order of the operations, new parallel communication can be constructed, where the loosely coupled local problems are nonlinear. We discuss different nonlinear FETI-DP methods which are equivalent when applied to linear problems but which show a different performance for nonlinear problems. Moreover, a new unified framework is introduced which casts all nonlinear FETI-DP domain decomposition approaches discussed in the literature into a single algorithm. Furthermore, the equivalence of nonlinear FETI-DP methods to specific nonlinearly right-preconditioned Newton-Krylov methods is shown. For the methods using nested Newton iterations, a strategy is presented to stop the inner Newton iteration early, resulting in an approximate local nonlinear elimination. Additionally, the nonlinear BDDC (Balancing Domain Decomposition by Constraint) method is presented as a right-preconditioned Newton approach. Finally, for the first time, parallel weak scaling results for four different nonlinear FETI-DP approaches are compared to standard Newton-Krylov FETI-DP in two and three dimensions, using both exact as well as highly scalable inexact linear FETI-DP preconditioners and up to 131 072 message passing interface (MPI) ranks on the JUQUEEN supercomputer at Forschungszentrum Julich. For a model problem with nonlocal nonlinearities, nonlinear FETI-DP methods are shown to be up to five times faster than the standard Newton-Krylov FETI-DP approach.

Parallel Newton-Krylov FETI-DP (Finite Element Tearing and Interconnecting-Dual-Primal) domain decomposition methods are fast and robust solvers, e.g., for nonlinear implicit problems in structural mechanics. In these methods, the nonlinear problem is first linearized and then decomposed into loosely coupled (linear) problems, which can be solved in parallel. By changing the order of the operations, new parallel communication can be constructed, where the loosely coupled local problems are nonlinear. We discuss different nonlinear FETI-DP methods which are equivalent when applied to linear problems but which show a different performance for nonlinear problems. Moreover, a new unified framework is introduced which casts all nonlinear FETI-DP domain decomposition approaches discussed in the literature into a single algorithm. Furthermore, the equivalence of nonlinear FETI-DP methods to specific nonlinearly right-preconditioned Newton-Krylov methods is shown. For the methods using nested Newton iterations, a strategy is presented to stop the inner Newton iteration early, resulting in an approximate local nonlinear elimination. Additionally, the nonlinear BDDC (Balancing Domain Decomposition by Constraint) method is presented as a right-preconditioned Newton approach. Finally, for the first time, parallel weak scaling results for four different nonlinear FETI-DP approaches are compared to standard Newton-Krylov FETI-DP in two and three dimensions, using both exact as well as highly scalable inexact linear FETI-DP preconditioners and up to 131 072 message passing interface (MPI) ranks on the JUQUEEN supercomputer at Forschungszentrum Julich. For a model problem with nonlocal nonlinearities, nonlinear FETI-DP methods are shown to be up to five times faster than the standard Newton-Krylov FETI-DP approach.

We describe a new implementation of a two-level overlapping Schwarz preconditioner with energy-minimizing coarse space (GDSW: generalized Dryja-Smith-Widlund) and show numerical results for an additive and a hybrid additive-multiplicative version. Our parallel implementation makes use of the Trilinos software library and provides a framework for parallel two-level Schwarz methods. We show parallel scalability for two- and three-dimensional scalar second-order elliptic and linear elasticity problems for several thousands of cores. We also discuss techniques for the parallel construction of coarse spaces which are also of interest for other parallel preconditioners and discretization methods using energy minimizing coarse functions. We finally show an application in monolithic fluid-structure interaction, where significant improvements are achieved compared to a standard algebraic, one-level overlapping Schwarz method.

An adaptive coarse space approach including a condition number bound for dual primal finite element tearing and interconnecting (FETI-DP) methods applied to three dimensional problems with coefficient jumps inside subdomains and across subdomain boundaries is presented. The approach is based on a known adaptive coarse space approach enriched by a small number of additional local edge eigenvalue problems. These edge eigenvalue problems serve to make the method robust and permit a condition number bound which depends only on the tolerance of the local eigenvalue problems and some properties of the domain decomposition. The introduction of the edge eigenvalue problems thus turns a well-known condition number indicator for FETI-DP and balancing domain decomposition by constraints (BDDC) methods into a condition number estimate. Numerical results are presented for linear elasticity and heterogeneous materials supporting our theoretical findings. The problems considered include those with random coefficients and almost incompressible material components.

An adaptive coarse space approach including a condition number bound for dual primal finite element tearing and interconnecting (FETI-DP) methods applied to three dimensional problems with coefficient jumps inside subdomains and across subdomain boundaries is presented. The approach is based on a known adaptive coarse space approach enriched by a small number of additional local edge eigenvalue problems. These edge eigenvalue problems serve to make the method robust and permit a condition number bound which depends only on the tolerance of the local eigenvalue problems and some properties of the domain decomposition. The introduction of the edge eigenvalue problems thus turns a well-known condition number indicator for FETI-DP and balancing domain decomposition by constraints (BDDC) methods into a condition number estimate. Numerical results are presented for linear elasticity and heterogeneous materials supporting our theoretical findings. The problems considered include those with random coefficients and almost incompressible material components.

We describe a new implementation of a two-level overlapping Schwarz preconditioner with energy-minimizing coarse space (GDSW: generalized Dryja-Smith-Widlund) and show numerical results for an additive and a hybrid additive-multiplicative version. Our parallel implementation makes use of the Trilinos software library and provides a framework for parallel two-level Schwarz methods. We show parallel scalability for two- and three-dimensional scalar second-order elliptic and linear elasticity problems for several thousands of cores. We also discuss techniques for the parallel construction of coarse spaces which are also of interest for other parallel preconditioners and discretization methods using energy minimizing coarse functions. We finally show an application in monolithic fluid-structure interaction, where significant improvements are achieved compared to a standard algebraic, one-level overlapping Schwarz method.

The solution of nonlinear problems, e.g., in material science, requires fast and highly scalable parallel solvers. Finite element tearing and interconnecting dual primal (FETI-DP) domain decomposition methods are parallel solution methods for implicit problems discretized by finite elements. Recently, nonlinear versions of the well-known FETI-DP methods for linear problems have been introduced. In these methods, the nonlinear problem is decomposed before linearization. This approach can be viewed as a strategy to further localize computational work and to extend the parallel scalability of FETI-DP methods toward extreme-scale supercomputers. Here, a recent nonlinear FETI-DP method is combined with an approach that allows an inexact solution of the FETI-DP coarse problem. We combine the nonlinear FETI-DP domain decomposition method with an algebraic multigrid (AMG) method and thus obtain a hybrid nonlinear domain decomposition/multigrid method. We consider scalar nonlinear problems as well as nonlinear hyperelasticity problems in two and three space dimensions. For the first time for a domain decomposition method, weak parallel scalability can be shown beyond half a million cores and subdomains. We can show weak parallel scalability for up to 524 288 cores on the Mira Blue Gene/Q supercomputer for our new implementation and discuss the steps necessary to obtain these results. We solve a heterogeneous nonlinear hyperelasticity problem discretized using piecewise quadratic finite elements with a total of 42 billion degrees of freedom in about six minutes. Our analysis reveals that scalability beyond 524 288 cores depends critically on both efficient construction and solution of the coarse problem.

The solution of nonlinear problems, e.g., in material science, requires fast and highly scalable parallel solvers. Finite element tearing and interconnecting dual primal (FETI-DP) domain decomposition methods are parallel solution methods for implicit problems discretized by finite elements. Recently, nonlinear versions of the well-known FETI-DP methods for linear problems have been introduced. In these methods, the nonlinear problem is decomposed before linearization. This approach can be viewed as a strategy to further localize computational work and to extend the parallel scalability of FETI-DP methods toward extreme-scale supercomputers. Here, a recent nonlinear FETI-DP method is combined with an approach that allows an inexact solution of the FETI-DP coarse problem. We combine the nonlinear FETI-DP domain decomposition method with an algebraic multigrid (AMG) method and thus obtain a hybrid nonlinear domain decomposition/multigrid method. We consider scalar nonlinear problems as well as nonlinear hyperelasticity problems in two and three space dimensions. For the first time for a domain decomposition method, weak parallel scalability can be shown beyond half a million cores and subdomains. We can show weak parallel scalability for up to 524 288 cores on the Mira Blue Gene/Q supercomputer for our new implementation and discuss the steps necessary to obtain these results. We solve a heterogeneous nonlinear hyperelasticity problem discretized using piecewise quadratic finite elements with a total of 42 billion degrees of freedom in about six minutes. Our analysis reveals that scalability beyond 524 288 cores depends critically on both efficient construction and solution of the coarse problem.

New nonlinear FETI-DP (dual-primal finite element tearing and interconnecting) and BDDC (balancing domain decomposition by constraints) domain decomposition methods are introduced. In all these methods, in each iteration, local nonlinear problems are solved on the subdomains. The new approaches can significantly reduce communication and show a significantly improved performance, especially for problems with localized nonlinearities, compared to a standard Newton-Krylov-FETI-DP or BDDC approach. Moreover, the coarse space of the nonlinear FETI-DP methods can be used to accelerate the Newton convergence. It is also found that the new nonlinear FETI-DP and nonlinear BDDC methods are not as closely related as in the linear context. Numerical results for the p-Laplace operator are presented.

New nonlinear FETI-DP (dual-primal finite element tearing and interconnecting) and BDDC (balancing domain decomposition by constraints) domain decomposition methods are introduced. In all these methods, in each iteration, local nonlinear problems are solved on the subdomains. The new approaches can significantly reduce communication and show a significantly improved performance, especially for problems with localized nonlinearities, compared to a standard Newton-Krylov-FETI-DP or BDDC approach. Moreover, the coarse space of the nonlinear FETI-DP methods can be used to accelerate the Newton convergence. It is also found that the new nonlinear FETI-DP and nonlinear BDDC methods are not as closely related as in the linear context. Numerical results for the p-Laplace operator are presented.