arXiv subjects

Stefan Takacs

Publications and source records attributed to Stefan Takacs.

At least 19 recordsLinked to original sources

arXiv2026

Robust approximation error estimates for analysis-suitable $G^1$ isogeometric multi-patch discretizations

We prove $p$-robust approximation error estimates for $H^2$-conforming isogeometric discretizations over planar multi-patch domains. Possible applications are fourth order boundary value problems, like the biharmonic equation or Kirchhoff-Love plates. Using Isogeometric Analysis, such conforming discretizations can be constructed effortlessly for the single-patch case. In order to obtain a globally $H^2$-conforming discretization in the multi-patch case, the functions must be $C^1$-smooth across the interfaces between the patches. To obtain optimal approximation properties, those $C^1$-smooth spaces must also reproduce splines of sufficiently high degree for traces and transversal derivatives at all patch interfaces. Such constructions are based on some assumptions on the geometry. We restrict ourselves to the class of analysis-suitable $G^1$ (AS-$G^1$) multi-patch domains, which is the subset of $C^0$-matching multi-patch domains that allows the definition of spline spaces that yield the necessary reproduction properties without the need to locally increase the degree. While approximation error estimates have been established for single-patch and $C^0$ isogeometric multi-patch spaces, corresponding results for the $C^1$ multi-patch setting have been missing. The resulting bounds on the approximation error depend on the geometry parameterization and on the Sobolev regularity of the target function, but are independent of the spline degree $p$.

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arXiv2025

The Isogeometric Fast Fourier-based Diagonalization method

The construction of robust solvers for linear systems obtained from the discretization of partial differential equations using Isogeometric Analysis is challenging since the condition number of the system matrix not only grows with the reciprocal square of the grid size (for second order problems), but also exponentially with the spline degree. The Fast Diagonalization method allows the construction of a preconditioner that is robust both in grid size and spline degree. Although this method is efficient in practice, its computational complexity is superlinear in the number of degrees of freedom. In this work, we construct a variant of the Fast Diagonalization method that can exploit the Fast Fourier Transformation. Note that, because of boundary effects, a Fourier Transformation cannot diagonalize the overall problem. We circumvent this issue using a stable splitting of the spline space. The resulting preconditioner is still robust with respect to the grid size and spline degree and can be realized with a computational complexity that grows almost linearly with the number of degrees of freedom.

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arXiv2025

An Isogeometric Tearing and Interconnecting method for conforming discretizations of the biharmonic problem

We propose and analyze a domain decomposition solver for the biharmonic problem. The problem is discretized in a conforming way using multi-patch Isogeometric Analysis. As first step, we discuss the setup of a sufficiently smooth discretization space. We focus on two dimensional computational domains that are parameterized with sufficiently smooth geometry functions. As solution technique, we use a variant of the Dual-Primal Finite Element Tearing and Interconnecting method that is also known as Dual-Primal Isogeometric Tearing and Interconnecting (IETI-DP) method in the context of Isogeometric Analysis. We present a condition number estimate and illustrate the behavior of the proposed method with numerical results.

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arXiv2025

Dual-primal Isogeometric Tearing and Interconnecting Solvers for adaptively refined multi-patch configurations

Isogeometric Analysis is a variant of the finite element method, where spline functions are used for the representation of both the geometry and the solution. Splines, particularly those with higher degree, achieve their full approximation power only if the solution is sufficiently regular. Since solutions are usually not regular everywhere, adaptive refinement is essential. Recently, a multi-patch-based adaptive refinement strategy based on recursive patch splitting has been proposed, which naturally generates hierarchical, non-matching multi-patch configurations with T-junctions, but preserves the tensor-product structure within each patch. In this work, we investigate the application of the dual-primal Isogeometric Tearing and Interconnecting method (IETI-DP) to such adaptive multi-patch geometries. We provide sufficient conditions for the solvability of the local problems and propose a preconditioner for the overall iterative solver. We establish a condition number bound that coincides with the bound previously shown for the fully matching case. Numerical experiments confirm the theoretical findings and demonstrate the efficiency of the proposed approach in adaptive refinement scenarios.

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arXiv2024

A robust and time-parallel preconditioner for parabolic reconstruction problems using Isogeometric Analysis

We consider a PDE-constrained optimization problem of tracking type with parabolic state equation. The solution to the problem is characterized by the Karush-Kuhn-Tucker (KKT) system, which we formulate using a strong variational formulation of the state equation and a super weak formulation of the adjoined state equation. This allows us to propose a preconditioner that is robust both in the regularization and the diffusion parameter. In order to discretize the problem, we use Isogeometric Analysis since it allows the construction of sufficiently smooth basis functions effortlessly. To realize the preconditioner, one has to solve a problem over the whole space time cylinder that is elliptic with respect to certain non-standard norms. Using a fast diagonalization approach in time, we reformulate the problem as a collection of elliptic problems in space only. These problems are not only smaller, but our approach also allows to solve them in a time-parallel way. We show the efficiency of the preconditioner by rigorous analysis and illustrate it with numerical experiments.

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arXiv2024

Multi-resolution Isogeometric Analysis -- Efficient adaptivity utilizing the multi-patch structure

Isogeometric Analysis (IgA) is a spline based approach to the numerical solution of partial differential equations. There are two major issues that IgA was designed to address. The first issue is the exact representation of domains stemming from Computer Aided Design (CAD) software. In practice, this can be realized only with multi-patch IgA, often in combination with trimming or similar techniques. The second issue is the realization of high-order discretizations (by increasing the spline degree) with numbers of degrees of freedom comparable to low-order methods. High-order methods can deliver their full potential only if the solution to be approximated is sufficiently smooth; otherwise, adaptive methods are required. In the last decades, a zoo of local refinement strategies for splines has been developed. The authors think that many of these approaches are a burden to implement efficiently and impede the utilization of recent advances that rely on tensor-product splines, e.g., concerning matrix assembly and preconditioning. The implementation seems to be particularly cumbersome in the context of multi-patch IgA. Our approach is to moderately increase the number of patches and to utilize different grid sizes on different patches. This allows reusing the existing code bases, recovers the convergence rates of other adaptive approaches and increases the number of degrees of freedom only marginally.

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arXiv2023

Isogeometric Tearing and Interconnecting Solvers for Linearized Elasticity in multi-patch Isogeometric Analysis

We consider the linearized elasticity equation, discretized with multi-patch Isogeometric Analysis. A standard discretization error analysis is based on Korn's inequality, which degrades for certain geometries, such as long and thin cantilevers. This phenomenon is known as geometry locking. We observe that high-order methods, like Isogeometric Analysis is beneficial in such a setting. The main focus of this paper is the construction and analysis of a domain decomposition solver, namely an Isogeometric Tearing and Interconnecting (IETI) solver, where we prove that the convergence behavior does not depend on the constant of Korn's inequality for the overall domain, but only on the corresponding constants for the individual patches. Moreover, our analysis is explicit in the choice of the spline degree. Numerical experiments are provided which demonstrates the efficiency of the proposed solver.

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arXiv2022

Stable discretizations and IETI-DP solvers for the Stokes system in multi-patch Isogeometric Analysis

We are interested in a fast solver for the Stokes equations, discretized with multi-patch Isogeometric Analysis. In the last years, several inf-sup stable discretizations for the Stokes problem have been proposed, often the analysis was restricted to single-patch domains. We focus on one of the simplest approaches, the isogeometric Taylor--Hood element. We show how stability results for single-patch domains can be carried over to multi-patch domains. While this is possible, the stability strongly depends on the shape of the geometry. We construct a Dual-Primal Isogeometric Tearing and Interconnecting (IETI-DP) solver that does not suffer from that effect. We give a convergence analysis and provide numerical tests.

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arXiv2022

A IETI-DP method for discontinuous Galerkin discretizations in Isogeometric Analysis with inexact local solvers

We construct solvers for an isogeometric multi-patch discretization, where the patches are coupled via a discontinuous Galerkin approach, which allows the consideration of discretizations that do not match on the interfaces. We solve the resulting linear system using a Dual-Primal IsogEometric Tearing and Interconnecting (IETI-DP) method. We are interested in solving the arising patch-local problems using iterative solvers since this allows the reduction of the memory footprint. We solve the patch-local problems approximately using the Fast Diagonalization method, which is known to be robust in the grid size and the spline degree. To obtain the tensor structure needed for the application of the Fast Diagonalization method, we introduce an orthogonal splitting of the local function spaces. We present a convergence theory that confirms that the condition number of the preconditioned system only grows poly-logarithmically with the grid size. The numerical experiments confirm this finding. Moreover, they show that the convergence of the overall solver only mildly depends on the spline degree. We observe a mild reduction of the computational times and a significant reduction of the memory requirements in comparison to standard IETI-DP solvers using sparse direct solvers for the local subproblems. Furthermore, the experiments indicate good scaling behavior on distributed memory machines.

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arXiv2021

Multigrid solvers for isogeometric discretizations of the second biharmonic problem

We develop a multigrid solver for the second biharmonic problem in the context of Isogeometric Analysis (IgA), where we also allow a zero-order term. In a previous paper, the authors have developed an analysis for the first biharmonic problem based on Hackbusch's framework. This analysis can only be extended to the second biharmonic problem if one assumes uniform grids. In this paper, we prove a multigrid convergence estimate using Bramble's framework for multigrid analysis without regularity assumptions. We show that the bound for the convergence rate is independent of the scaling of the zero-order term and the spline degree. It only depends linearly on the number of levels, thus logarithmically on the grid size. Numerical experiments are provided which illustrate the convergence theory and the efficiency of the proposed multigrid approaches.

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arXiv2021

Dual-Primal Isogeometric Tearing and Interconnecting methods for the Stokes problem

We are interested in a fast solver for linear systems obtained by discretizing the Stokes problem with multi-patch Isogeometric Analysis. We use Dual-Primal Isogeometric Tearing and Interconnecting (IETI-DP) methods. In resent years, IETI-DPand related methods have been studied extensively, mainly for the Poisson problem. For the Stokes equations, several challenges arise since the corresponding system is not positive definite, but has saddle point structure. Moreover, the Stokes equations with Dirichlet boundary conditions have a null-space, consisting of the constant pressure modes. This poses a challenge when considering the scaled Dirichlet preconditioner. We test out two different scaled Dirichlet preconditioners with different choices of primal degrees of freedom. The tests are performed on rather simple domains (the unit square and a quarter annulus) and a more complicated domain (a Yeti-footprint).

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arXiv2021

Inexact IETI-DP for conforming isogeometric multi-patch discretizations

In this paper, we investigate Dual-Primal Isogeometric Tearing and Interconnecting (IETI-DP) methods for conforming Galerkin discretizations on multi-patch computational domains with inexact subdomain solvers. Recently, the authors have proven a condition number estimate for a IETI-DP method using sparse LU factorizations for the subdomain problems that is explicit, among other parameters, in the grid size and the spline degree. In the present paper, we replace the sparse LU factorizations by fast diagonalization based preconditioners to get a faster IETI-DP method while maintaining the same explicit condition number bound.

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arXiv2021

IETI-DP methods for discontinuous Galerkin multi-patch Isogeometric Analysis with T-junctions

We study Dual-Primal Isogeometric Tearing and Interconnecting (IETI-DP) solvers for non-conforming multi-patch discretizations of a generalized Poisson problem. We realize the coupling between the patches using a symmetric interior penalty discontinuous Galerkin (SIPG) approach. Previously, we have assumed that the interfaces between patches always consist of whole edges. In this paper, we drop this requirement and allow T-junctions. This extension is vital for the consideration of sliding interfaces, for example between the rotor and the stator of an electrical motor. One critical part for the handling of T-junctions in IETI-DP solvers is the choice of the primal degrees of freedom. We propose to add all basis functions that are non-zero at any of the vertices to the primal space. Since there are several such basis functions at any T-junction, we call this concept ''fat vertices''. For this choice, we show a condition number bound that coincides with the bound for the conforming cas

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arXiv2021

A quasi-robust discretization error estimate for discontinuous Galerkin Isogeometric Analysis

Isogeometric Analysis is a spline-based discretization method to partial differential equations which shows the approximation power of a high-order method. The number of degrees of freedom, however, is as small as the number of degrees of freedom of a low-order method. This does not come for free as the original formulation of Isogeometric Analysis requires a global geometry function. Since this is too restrictive for many kinds of applications, the domain is usually decomposed into patches, where each patch is parameterized with its own geometry function. In simpler cases, the patches can be combined in a conforming way. However, for non-matching discretizations or for varying coefficients, a non-conforming discretization is desired. An symmetric interior penalty discontinuous Galerkin (SIPG) method for Isogeometric Analysis has been previously introduced. In the present paper, we give error estimates that only depend poly-logarithmically on the spline degree. This opens the door towards the construction and the analysis of fast linear solvers, particularly multigrid solvers for non-conforming multipatch Isogeometric Analysis.

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arXiv2021

IETI-DP for conforming multi-patch Isogeometric Analysis in three dimensions

We consider dual-primal isogeometric tearing and interconnection (IETI-DP) solvers for multi-patch geometries in Isogeometric Analysis. Recently, the authors have published a convergence analysis for those solvers that is explicit in both the grid size and the spline degree for conforming discretizations of two dimensional computational domains. In the present paper, we shortly revisit these results and provide numerical experiments that indicate that similar results may hold for three dimensional domains.

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arXiv2021

Towards a IETI-DP solver on non-matching multi-patch domains

Recently, the authors have proposed and analyzed isogeometric tearing and interconnecting (IETI-DP) solvers for multi-patch discretizations in Isogeometric Analysis. Conforming and discontinuous Galerkin settings have been considered. In both cases, we have assumed that the interfaces between the patches consist of whole edges. In this paper, we present a generalization that allows us to drop this requirement. This means that the patches can meet in T-junctions, which increases the flexibility of the geometric model significantly. We use vertex-based primal degrees of freedom. For the T-junctions, we propose to follow the idea of "fat vertices".

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arXiv2020

Robust preconditioning and error estimates for optimal control of the convection-diffusion-reaction equation with limited observation in Isogeometric analysis

In this paper we analyze an optimization problem with limited observation governed by a convection--diffusion--reaction equation. Motivated by a Schur complement approach, we arrive at continuous norms that enable analysis of well-posedness and subsequent derivation of error analysis and a preconditioner that is robust with respect to the parameters of the problem. We provide conditions for inf-sup stable discretizations and present one such discretization for box domains with constant convection. We also provide a priori error estimates for this discretization. The preconditioner requires a fourth order problem to be solved. For this reason, we use Isogeometric Analysis as a method of discretization. To efficiently realize the preconditioner, we consider geometric multigrid with a standard Gauss-Seidel smoother as well as a new macro Gauss-Seidel smoother. The latter smoother provides good results with respect to both the geometry mapping and the polynomial degree.

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arXiv2020

Convergence theory for IETI-DP solvers for discontinuous Galerkin Isogeometric Analysis that is explicit in h and p

In this paper, we develop a convergence theory for Dual-Primal Isogeometric Tearing and Interconnecting (IETI-DP) solvers for isogeometric multi-patch discretizations of the Poisson problem, where the patches are coupled using discontinuous Galerkin. The presented theory provides condition number bounds that are explicit in the grid sizes h and in the spline degrees p. We give an analysis that holds for various choices for the primal degrees of freedom: vertex values, edge averages, and a combination of both. If only the vertex values or both vertex values and edge averages are taken as primal degrees of freedom, the condition number bound is the same as for the conforming case. If only the edge averages are taken, both the convergence theory and the experiments show that the condition number of the preconditioned system grows with the ratio of the grid sizes on neighboring patches.

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