Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Overlapping subspaces and singular systems with application to Isogeometric Analysis

Andrea Bressan, Massimiliano Martinelli, Giancarlo Sangalli

TL;DR

A framework for solving partial differential equations (PDEs) motivated by isogeometric analysis (IGA) and local tensor-product splines is proposed, which leads to a potentially singular linear system, which is handled by a Krylov linear solver.

Abstract

We propose a framework for solving partial differential equations (PDEs) motivated by isogeometric analysis (IGA) and local tensor-product splines. Instead of using a global basis for the solution space we use as generators the disjoint union of subspace bases. This leads to a potentially singular linear system, which is handled by a Krylov linear solver. The framework may offer computational advantages in dealing with spaces like Hierarchical B-splines, T-splines, and LR-splines.

Overlapping subspaces and singular systems with application to Isogeometric Analysis

TL;DR

A framework for solving partial differential equations (PDEs) motivated by isogeometric analysis (IGA) and local tensor-product splines is proposed, which leads to a potentially singular linear system, which is handled by a Krylov linear solver.

Abstract

We propose a framework for solving partial differential equations (PDEs) motivated by isogeometric analysis (IGA) and local tensor-product splines. Instead of using a global basis for the solution space we use as generators the disjoint union of subspace bases. This leads to a potentially singular linear system, which is handled by a Krylov linear solver. The framework may offer computational advantages in dealing with spaces like Hierarchical B-splines, T-splines, and LR-splines.
Paper Structure (8 sections, 5 theorems, 45 equations, 2 figures, 2 tables)

This paper contains 8 sections, 5 theorems, 45 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Abstract setting
  3. Krylov methods
  4. CG and MINRES
  5. GMRES
  6. Subspace correction methods
  7. Numerical examples
  8. Conclusions

Key Result

Theorem 1

If $A:V\rightarrow V^*$ is an isomorphism, then and

Figures (2)

  • Figure 1: 2D case: the physical domain $F(\widehat{\Omega}_0)$ associated to the space $V_0$ for $L=0$.
  • Figure 2: 2D case: the different physical domains associated to the spaces $V_{1}^x$ (\ref{['fig:2d_V1x']}), $V_{2}^x$ (\ref{['fig:2d_V2x']}), $V_{1}^y$ (\ref{['fig:2d_V1y']}) and $V_{2}^y$ (\ref{['fig:2d_V2y']}) for $L=0$.

Theorems & Definitions (10)

  • Theorem 1
  • proof
  • Proposition 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • proof