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Space-Time Isogeometric Method for a Nonlocal Parabolic Problem

Sudhakar Chaudhary, Shreya Chauhan, Monica Montardini

TL;DR

The paper tackles a nonlinear nonlocal parabolic PDE with diffusion depending on a global quantity $l(u)=\int_{\Omega} u$. It develops a space-time isogeometric discretization, proves existence and uniqueness of the continuous weak solution and existence of a discrete solution, and establishes a priori error estimates in the $W$-norm using a Céa-type argument. A Picard linearization combined with a specialized preconditioned GMRES solver is used to solve the resulting systems efficiently, leveraging a time-pencil factorization and tensor-product IgA operators. Numerical experiments on multiple geometric domains demonstrate optimal convergence rates, robustness of the preconditioner with respect to meshsize and polynomial degree, and the superior accuracy of the space-time approach over traditional time-stepping. Collectively, the work validates space-time IgA as a viable and high-accuracy method for nonlinear nonlocal parabolic problems, with potential for parallel-time solutions and improved geometric exactness.

Abstract

In the present work, we focus on the space-time isogeometric discretization of a parabolic problem with a nonlocal diffusion coefficient. The existence and uniqueness of the solution for the continuous space-time variational formulation are proven. We prove the existence of the discrete solution and also establish the a priori error estimate for the space-time isogeometric scheme. The non-linear system is linearized through Picards method and a suitable preconditioner for the linearized system is provided. Finally, to confirm the theoretical findings, results of some numerical experiments are presented.

Space-Time Isogeometric Method for a Nonlocal Parabolic Problem

TL;DR

The paper tackles a nonlinear nonlocal parabolic PDE with diffusion depending on a global quantity . It develops a space-time isogeometric discretization, proves existence and uniqueness of the continuous weak solution and existence of a discrete solution, and establishes a priori error estimates in the -norm using a Céa-type argument. A Picard linearization combined with a specialized preconditioned GMRES solver is used to solve the resulting systems efficiently, leveraging a time-pencil factorization and tensor-product IgA operators. Numerical experiments on multiple geometric domains demonstrate optimal convergence rates, robustness of the preconditioner with respect to meshsize and polynomial degree, and the superior accuracy of the space-time approach over traditional time-stepping. Collectively, the work validates space-time IgA as a viable and high-accuracy method for nonlinear nonlocal parabolic problems, with potential for parallel-time solutions and improved geometric exactness.

Abstract

In the present work, we focus on the space-time isogeometric discretization of a parabolic problem with a nonlocal diffusion coefficient. The existence and uniqueness of the solution for the continuous space-time variational formulation are proven. We prove the existence of the discrete solution and also establish the a priori error estimate for the space-time isogeometric scheme. The non-linear system is linearized through Picards method and a suitable preconditioner for the linearized system is provided. Finally, to confirm the theoretical findings, results of some numerical experiments are presented.
Paper Structure (16 sections, 9 theorems, 90 equations, 7 figures, 2 tables, 2 algorithms)

This paper contains 16 sections, 9 theorems, 90 equations, 7 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Function spaces
  4. Isogeometric spaces
  5. Special types of operators
  6. Space-time weak formulation: existence-uniqueness results
  7. Space-time isogeometric discretization
  8. Discrete system and linear solver
  9. Convergence analysis
  10. Numerical experiments
  11. Orders of convergence: quarter of annulus domain
  12. Orders of convergence: thick ring-shaped domain
  13. Comparison of time-stepping and space-time method
  14. Performance of preconditioner: igloo-shaped domain
  15. A numerical experiment with different nonlocal term
  16. ...and 1 more sections

Key Result

Theorem 2.1

LOLI20202586 Let $r$ be an integer such that $1< r\leq \min\{q_s,q_t\}+1$ and $u\in V\cap H^r(\Omega\times(0,T))$. Then there exists a projection $\mathcal{P}_h: V\cap H^r(\Omega\times(0,T))\rightarrow V_h$ such that where the constant $C$ is independent of the meshsizes $h_s$ and $h_t$.

Figures (7)

  • Figure 1: Computational domains.
  • Figure 2: Errors in $L^2(0,T;L^2(\Omega))$ and $L^2(0,T;H^1_0(\Omega))$ norm for quarter of annulus domain.
  • Figure 3: Errors in $L^2(0,T;L^2(\Omega))$ and $L^2(0,T;H^1_0(\Omega))$ norm for the thick ring-shaped domain.
  • Figure 4: Errors in $L^2(0,T;L^2(\Omega))$ and $L^2(0,T;H^1_0(\Omega))$ norm for INC-CN and ST-IgA approaches
  • Figure 5: Errors in $L^2(0,T;L^2(\Omega))$ and $L^2(0,T;H^1_0(\Omega))$ norm for INC-CN and ST-IgA approaches
  • ...and 2 more figures

Theorems & Definitions (16)

  • Theorem 2.1
  • Definition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 3.1
  • proof
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • Theorem 4.1
  • ...and 6 more