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

Stefan Takacs

TL;DR

This paper addresses efficient solvers for the biharmonic problem $\Delta^2 u=f$ on a domain $\Omega$ decomposed into multiple IgA patches, enforcing $C^1$ continuity to achieve $H^2$-conformity. It develops a Dual-Primal Isogeometric Tearing and Interconnecting (IETI-DP) domain decomposition method with a patch-wise scaled Dirichlet preconditioner, integrated with an explicit construction of discrete biharmonic extensions to enable stable cross-patch communication. A rigorous condition-number estimate is established: $\kappa(MF) \le C p^3 \big(1+\log p + \max_k \log\tfrac{H_k}{h_k}\big)^2$, and this polylogarithmic behavior is corroborated by numerical tests on complex multi-patch geometries, including a quarter annulus and a D-shaped lamella with a hole. The results demonstrate that the proposed method is robust with respect to mesh refinement and spline degree, highlighting its practical viability for high-order IgA discretizations of fourth-order problems. Future work includes extending the analysis to analysis-suitable $G^1$ discretizations and more general geometries.

Abstract

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.

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

TL;DR

This paper addresses efficient solvers for the biharmonic problem on a domain decomposed into multiple IgA patches, enforcing continuity to achieve -conformity. It develops a Dual-Primal Isogeometric Tearing and Interconnecting (IETI-DP) domain decomposition method with a patch-wise scaled Dirichlet preconditioner, integrated with an explicit construction of discrete biharmonic extensions to enable stable cross-patch communication. A rigorous condition-number estimate is established: , and this polylogarithmic behavior is corroborated by numerical tests on complex multi-patch geometries, including a quarter annulus and a D-shaped lamella with a hole. The results demonstrate that the proposed method is robust with respect to mesh refinement and spline degree, highlighting its practical viability for high-order IgA discretizations of fourth-order problems. Future work includes extending the analysis to analysis-suitable discretizations and more general geometries.

Abstract

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.

Paper Structure

This paper contains 13 sections, 22 theorems, 128 equations, 6 figures, 6 tables.

Table of Contents

  1. Introduction
  2. The model problem and its discretization
  3. The setup of the IETI-DP solver
  4. Estimates for the discrete biharmonic extension
  5. Preliminary results
  6. Extension operator for parameter domain
  7. Extension operator on the physical domain
  8. The condition number estimate
  9. Numerical results
  10. B-spline quarter annulus
  11. D-shaped lamella with hole
  12. A modified preconditioner
  13. Conclusions

Key Result

Lemma 4.1

Let $p\ge3$ and $\Xi$ be a $p$-open knot vector such that $S_{p,\Xi}\subset H^2(0,1)$. The $H^2$-orthogonal projector $\Pi : H^2_0(0,1) \to S_{p,\Xi} \cap H^2_0(0,1)$ satisfies $| v|_{H^2(0,1)}^2 = | \Pi v|_{H^2(0,1)}^2 + | v-\Pi v|_{H^2(0,1)}^2$ and the error estimate

Figures (6)

  • Figure 1: Parametrization of two neighboring patches
  • Figure 2: Example of transformed basis
  • Figure 3: Constraints between two patches.
  • Figure 4: Constraints between four patches.
  • Figure 5: Annulus domain, corresponding control mesh, and solution field as obtained with $16$ patches, $r=5$ and $p=3$
  • ...and 1 more figures

Theorems & Definitions (24)

  • Remark 2.1
  • Remark 2.2
  • Lemma 4.1
  • Lemma 4.2
  • Lemma 4.3
  • Lemma 4.4
  • Lemma 4.5
  • Lemma 4.6
  • Lemma 4.7
  • Lemma 4.8
  • ...and 14 more