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An unconditionally stable space-time isogeometric method for a biharmonic wave equation

S. Chauhan, S. Chaudhary

Abstract

This work presents a space-time isogeometric analysis of biharmonic wave problem, in contrast to the more common application of space-time methods to second order wave equations. We first establish the unique solvability of the continuous space-time variational formulation. In order to obtain $H^2$- conforming discretization of the biharmonic wave equation, we consider globally smooth B-spline functions having continuity higher than $C^0$. We prove that the resulting space-time discrete formulation is stable under a Courant-Friedrichs-Lewy (CFL) condition. Furthermore, we propose a stabilized formulation, achieved by adding a non-consistent penalty term, which yields unconditional stability. Exploiting the tensor product structure, an efficient direct solver is also provided for solving the linear system arising from the discrete formulation. A few numerical experiments are presented to demonstrate the stability and convergence properties of the proposed scheme as well as the efficiency of the proposed solver.

An unconditionally stable space-time isogeometric method for a biharmonic wave equation

Abstract

This work presents a space-time isogeometric analysis of biharmonic wave problem, in contrast to the more common application of space-time methods to second order wave equations. We first establish the unique solvability of the continuous space-time variational formulation. In order to obtain - conforming discretization of the biharmonic wave equation, we consider globally smooth B-spline functions having continuity higher than . We prove that the resulting space-time discrete formulation is stable under a Courant-Friedrichs-Lewy (CFL) condition. Furthermore, we propose a stabilized formulation, achieved by adding a non-consistent penalty term, which yields unconditional stability. Exploiting the tensor product structure, an efficient direct solver is also provided for solving the linear system arising from the discrete formulation. A few numerical experiments are presented to demonstrate the stability and convergence properties of the proposed scheme as well as the efficiency of the proposed solver.

Paper Structure

This paper contains 8 sections, 4 theorems, 89 equations, 11 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries and foundational results
  3. Well-posedness of space-time variational formulation
  4. Space-time isogeometric discretization and stability analysis
  5. Stability analysis
  6. Efficient solver
  7. Numerical experiments
  8. Conclusions

Key Result

Theorem 2.1

coer Let $f\in (H^1_{,0}(0,T))'$. Then there exists a unique $\varphi\in H^1_{0,}(0,T)$ which satisfies odevfh and $\blacktriangleleft$$\blacktriangleleft$

Figures (11)

  • Figure 1: Computational domains.
  • Figure 2: Relative errors in (a) $L^2(L^2(\Omega))$ norm, (b) $L^2(H^1_0(\Omega))\cap H^1(L^2(\Omega))$ norm and (c) $X$ norm for the IgA stabilization method with splines of maximum regularity in both space and time direction for Square domain.
  • Figure 3: Relative errors in (a) $L^2(L^2(\Omega))$ norm, (b) $L^2(H^1_0(\Omega))\cap H^1(L^2(\Omega))$ norm and (c) $X$ norm for the IgA stabilization method with splines of maximum regularity in space and $C^{p-2}$ regularity in time direction for Square domain.
  • Figure 4: Relative errors in (a) $L^2(L^2(\Omega))$ norm, (b) $L^2(H^1_0(\Omega))\cap H^1(L^2(\Omega))$ norm and (c) $X$ norm for the IgA stabilization method with splines of maximum regularity in space and $C^0$ regularity in time direction for Square domain.
  • Figure 5: Relative errors in (a) $L^2(L^2(\Omega))$ norm, (b) $L^2(H^1_0(\Omega))\cap H^1(L^2(\Omega))$ norm and (c) $X$ norm for the FEM stabilization method with splines of maximum regularity in both space and time direction for Square domain.
  • ...and 6 more figures

Theorems & Definitions (8)

  • Theorem 2.1
  • Lemma 2.1
  • Theorem 2.2
  • Theorem 3.1
  • proof
  • Remark 4.1
  • Remark 4.2
  • Remark 5.1