arXiv subjects

Stefan Takacs

Publications and source records attributed to Stefan Takacs.

32 records · Page 2Linked to original sources

arXiv2019

Condition number bounds for IETI-DP methods that are explicit in h and p

We study the convergence behavior of Dual-Primal Isogeometric Tearing and Interconnecting (IETI-DP) methods for solving large-scale algebraic systems arising from multi-patch Isogeometric Analysis. We focus on the Poisson problem on two dimensional computational domains. We provide a convergence analysis that covers several choices of the primal degrees of freedom: the vertex values, the edge averages, and the combination of both. We derive condition number bounds that show the expected behavior in the grid size h and that are quasi-linear in the spline degree p.

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arXiv2019

Fast multigrid solvers for conforming and non-conforming multi-patch Isogeometric Analysis

Isogeometric Analysis is a high-order discretization method for boundary value problems that uses a number of degrees of freedom which is as small as for a low-order method. Standard isogeometric discretizations require a global parameterization of the computational domain. In non-trivial cases, the domain is decomposed into patches having separate parameterizations and separate discretization spaces. If the discretization spaces agree on the interfaces between the patches, the coupling can be done in a conforming way. Otherwise, non-conforming discretizations (utilizing discontinuous Galerkin approaches) are required. The author and his coworkers have previously introduced multigrid solvers for Isogeometric Analysis for the conforming case. In the present paper, these results are extended to the non-conforming case. Moreover, it is shown that the multigrid solves get even more powerful if the proposed smoother is combined with a (standard) Gauss-Seidel smoother.

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arXiv2018

A parallel multigrid solver for multi-patch Isogeometric Analysis

Isogeometric Analysis (IgA) is a framework for setting up spline-based discretizations of partial differential equations, which has been introduced around a decade ago and has gained much attention since then. If large spline degrees are considered, one obtains the approximation power of a high-order method, but the number of degrees of freedom behaves like for a low-order method. One important ingredient to use a discretization with large spline degree, is a robust and preferably parallelizable solver. While numerical evidence shows that multigrid solvers with standard smoothers (like Gauss Seidel) does not perform well if the spline degree is increased, the multigrid solvers proposed by the authors and their co-workers proved to behave optimal both in the grid size and the spline degree. In the present paper, the authors want to show that those solvers are parallelizable and that they scale well in a parallel environment.

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arXiv2018

Robust approximation error estimates and multigrid solvers for isogeometric multi-patch discretizations

In recent publications, the author and his coworkers have shown robust approximation error estimates for B-splines of maximum smoothness and have proposed multigrid methods based on them. These methods allow to solve the linear system arizing from the discretization of a partial differential equation in Isogeometric Analysis in a single-patch setting with convergence rates that are provably robust both in the grid size and the spline degree. In real-world problems, the computational domain cannot be nicely represented by just one patch. In computer aided design, such domains are typically represented as a union of multiple patches. In the present paper, we extend the approximation error estimates and the multigrid solver to this multi-patch case.

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arXiv2018

Robust multigrid solvers for the biharmonic problem in isogeometric analysis

In this paper, we develop multigrid solvers for the biharmonic problem in the framework of isogeometric analysis (IgA). In this framework, one typically sets up B-splines on the unit square or cube and transforms them to the domain of interest by a global smooth geometry function. With this approach, it is feasible to set up $H^2$-conforming discretizations. We propose two multigrid methods for such a discretization, one based on Gauss Seidel smoothing and one based on mass smoothing. We prove that both are robust in the grid size, the latter is also robust in the spline degree. Numerical experiments illustrate the convergence theory and indicate the efficiency of the proposed multigrid approaches, particularly of a hybrid approach combining both smoothers.

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arXiv2017

Inexact Dual-Primal Isogeometric Tearing and Interconnecting Methods

In this paper, we investigate inexact variants of dual-primal isogeometric tearing and interconnecting methods for solving large-scale systems of linear equations arising from Galerkin isogeometric discretizations of elliptic boundary value problems. The considered methods are extensions of standard finite element tearing and interconnecting methods to isogeometric analysis. The algorithms are implemented by means of energy minimizing primal subspaces. We discuss the replacement of local sparse direct solvers by iterative methods, particularly, multigrid solvers. We investigate the incorporation of these iterative solvers into different formulations of the algorithm. Finally, we present numerical examples comparing the performance of these inexact versions.

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arXiv2017

Robust multigrid methods for isogeometric discretizations of the Stokes equations

In recent publications, the author and his coworkers have proposed a multigrid method for solving linear systems arizing from the discretization of partial differential equations in isogeometric analysis and have proven that the convergence rates are robust in both the grid size and the polynomial degree. So, far the method has only been discussed for the Poisson problem. In the present paper, we want to face the question if it is possible to extend the method to the Stokes equations.

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arXiv2016

Robust Multigrid for Isogeometric Analysis Based on Stable Splittings of Spline Spaces

We present a robust and efficient multigrid method for single-patch isogeometric discretizations using tensor product B-splines of maximum smoothness. Our method is based on a stable splitting of the spline space into a large subspace of interior splines which satisfy a robust inverse inequality, as well as one or several smaller subspaces which capture the boundary effects responsible for the spectral outliers which occur in Isogeometric Analysis. We then construct a multigrid smoother based on an additive subspace correction approach, applying a different smoother to each of the subspaces. For the interior splines, we use a mass smoother, whereas the remaining components are treated with suitably chosen Kronecker product smoothers or direct solvers. We prove that the resulting multigrid method exhibits iteration numbers which are robust with respect to the spline degree and the mesh size. Furthermore, it can be efficiently realized for discretizations of problems in arbitrarily high geometric dimension. Some numerical examples illustrate the theoretical results and show that the iteration numbers also scale relatively mildly with the problem dimension.

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arXiv2016

Efficient smoothers for all-at-once multigrid methods for Poisson and Stokes control problems

In the present paper we concentrate on an important issue in constructing a good multigrid solver: the choice of an efficient smoother. We will introduce all-at-once multigrid solvers for optimal control problems which show robust convergence in the grid size and in the regularization parameter. We will refer to recent publications that guarantee such a convergence behavior. These publications do not pay much attention to the construction of the smoother and suggest to use a normal equation smoother. We will see that using a Gauss Seidel like variant of this smoother, the overall multigrid solver is speeded up by a factor of about two with no additional work. The author will give a proof which indicates that also the Gauss Seidel like variant of the smoother is covered by the convergence theory. Numerical experiments suggest that the proposed method are competitive with Vanka type methods.

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arXiv2016

A robust multigrid method for the time-dependent Stokes problem

In the present paper we propose a coupled multigrid method for generalized Stokes flow problems. Such problems occur as subproblems in implicit time-stepping approaches for time-dependent Stokes problems. The discretized Stokes system is a large-scale linear system whose condition number depends on the grid size of the spatial discretization and of the length of the time step. Recently, for this problem a coupled multigrid method has been proposed, where in each smoothing step a Poisson problem has to be solved (approximately) for the pressure field. In the present paper, we propose a coupled multigrid method where the solution of such sub-problems is not needed. We prove that the proposed method shows robust convergence behavior in the grid size of the spatial discretization and of the length of the time-step.

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arXiv2015

A Robust Multigrid Method for Isogeometric Analysis using Boundary Correction

We consider geometric multigrid methods for the solution of linear systems arising from isogeometric discretizations of elliptic partial differential equations. For classical finite elements, such methods are well known to be fast solvers showing optimal convergence behavior. However, the naive application of multigrid to the isogeometric case results in significant deterioration of the convergence rates if the spline degree is increased. Recently, a robust approximation error estimate and a corresponding inverse inequality for B-splines of maximum smoothness have been shown, both with constants independent of the spline degree. We use these results to construct multigrid solvers for discretizations based on B-splines with maximum smoothness which exhibit robust convergence rates.

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arXiv2015

Approximation error estimates and inverse inequalities for B-splines of maximum smoothness

In this paper, we develop approximation error estimates as well as corresponding inverse inequalities for B-splines of maximum smoothness, where both the function to be approximated and the approximation error are measured in standard Sobolev norms and semi-norms. The presented approximation error estimates do not depend on the polynomial degree of the splines but only on the grid size. We will see that the approximation lives in a subspace of the classical B-spline space. We show that for this subspace, there is an inverse inequality which is also independent of the polynomial degree. As the approximation error estimate and the inverse inequality show complementary behavior, the results shown in this paper can be used to construct fast iterative methods for solving problems arising from isogeometric discretizations of partial differential equations.

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arXiv2015

A robust all-at-once multigrid method for the Stokes control problem

In this paper we present an all-at-once multigrid method for a distributed Stokes control problem (velocity tracking problem). For solving such a problem, we use the fact that the solution is characterized by the optimality system (Karush-Kuhn-Tucker-system). The discretized optimality system is a large-scale linear system whose condition number depends on the grid size and on the choice of the regularization parameter forming a part of the problem. Recently, block-diagonal preconditioners have been proposed, which allow to solve the problem using a Krylov space method with convergence rates that are robust in both, the grid size and the regularization parameter or cost parameter. In the present paper, we develop an all-at-once multigrid method for a Stokes control problem and show robust convergence, more precisely, we show that the method converges with rates which are bounded away from one by a constant which is independent of the grid size and the choice of the regularization or cost parameter.

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arXiv2015

Using cylindrical algebraic decomposition and local Fourier analysis to study numerical methods: two examples

Local Fourier analysis is a strong and well-established tool for analyzing the convergence of numerical methods for partial differential equations. The key idea of local Fourier analysis is to represent the occurring functions in terms of a Fourier series and to use this representation to study certain properties of the particular numerical method, like the convergence rate or an error estimate. In the process of applying a local Fourier analysis, it is typically necessary to determine the supremum of a more or less complicated term with respect to all frequencies and, potentially, other variables. The problem of computing such a supremum can be rewritten as a quantifier elimination problem, which can be solved with cylindrical algebraic decomposition, a well-known tool from symbolic computation. The combination of local Fourier analysis and cylindrical algebraic decomposition is a machinery that can be applied to a wide class of problems. In the present paper, we will discuss two examples. The first example is to compute the convergence rate of a multigrid method. As second example we will see that the machinery can also be used to do something rather different: We will compare approximation error estimates for different kinds of discretizations.

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