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Space-time isogeometric method for a linear fourth order time dependent problem

Shreya Chauhan, Sudhakar Chaudhary

TL;DR

This work develops a space-time isogeometric analysis for a linear time-dependent fourth-order PDE by introducing an auxiliary variable to split the problem into a system of second-order equations and discretizing with tensor-product B-spline spaces. The authors establish well-posedness of the continuous space-time variational formulation via inf-sup stability and injectivity, and derive discrete inf-sup stability to obtain a Céa-type error bound. They formulate the discrete system with a Kronecker-product structure, proving convergence rates that align with the approximation properties of IgA spaces, and validate the theory through numerical tests on convex and non-convex domains. The study highlights the potential of space-time IgA for simultaneous space-time refinement of high-order parabolic problems and outlines future work on nonlinear extensions and solver efficiency.

Abstract

This article focuses on the space-time isogeometric method for a linear time dependent fourth order problem. Using an auxiliary variable, first the problem is split into a system of two second order differential equations and then the system is discretized by employing the tensor product spline spaces of time and spatial variables. We use the Babuska's theorem to prove the well-posedness of the continuous variational formulation. Also, the inf-sup stability condition at discrete level is established, which we use to prove the error estimates for the proposed method. Finally, to demonstrate the convergence of the scheme, few numerical results are reported.

Space-time isogeometric method for a linear fourth order time dependent problem

TL;DR

This work develops a space-time isogeometric analysis for a linear time-dependent fourth-order PDE by introducing an auxiliary variable to split the problem into a system of second-order equations and discretizing with tensor-product B-spline spaces. The authors establish well-posedness of the continuous space-time variational formulation via inf-sup stability and injectivity, and derive discrete inf-sup stability to obtain a Céa-type error bound. They formulate the discrete system with a Kronecker-product structure, proving convergence rates that align with the approximation properties of IgA spaces, and validate the theory through numerical tests on convex and non-convex domains. The study highlights the potential of space-time IgA for simultaneous space-time refinement of high-order parabolic problems and outlines future work on nonlinear extensions and solver efficiency.

Abstract

This article focuses on the space-time isogeometric method for a linear time dependent fourth order problem. Using an auxiliary variable, first the problem is split into a system of two second order differential equations and then the system is discretized by employing the tensor product spline spaces of time and spatial variables. We use the Babuska's theorem to prove the well-posedness of the continuous variational formulation. Also, the inf-sup stability condition at discrete level is established, which we use to prove the error estimates for the proposed method. Finally, to demonstrate the convergence of the scheme, few numerical results are reported.
Paper Structure (8 sections, 9 theorems, 75 equations, 5 figures)

This paper contains 8 sections, 9 theorems, 75 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Space-time weak formulation and well-posedness results
  3. Discrete formulation and error analysis
  4. B-spline spaces
  5. Discrete system
  6. Error analysis
  7. Numerical results
  8. Conclusions

Key Result

Theorem 2.1

Ern2004 Let $Z$ be a Banach space and $U$ be a reflexive Banach space. Let $a:Z\times U\rightarrow\mathbb{R}$ be a bounded bilinear form and $f\in U^*$. Then there exists a unique $u\in Z$ satisfying $a(u,v)=f(v) \ \forall v\in U$ if and only if the following two conditions hold: Moreover, the solution $u$ satisfies $\vert\vert u\vert\vert_Z\leq \frac{1}{\beta}\vert\vert f\vert\vert_{U^*}$.

Figures (5)

  • Figure 1: Computational domains.
  • Figure 2: Relative errors $E_u^1$ and $E_u^2$ for square domain.
  • Figure 3: Relative errors $E_v^1$ and $E_v^2$ for square domain.
  • Figure 4: Relative errors $E_u^1$ and $E_u^2$ for ring shaped domain.
  • Figure 5: Relative errors $E_v^1$ and $E_v^2$ for ring shaped domain.

Theorems & Definitions (15)

  • Theorem 2.1
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Theorem 2.2
  • Remark 2.1
  • Lemma 3.1
  • proof
  • ...and 5 more