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

TL;DR

This paper introduces a stable space-time isogeometric method for the biharmonic wave equation, featuring a stabilized formulation that achieves unconditional stability and an efficient solver, demonstrated through numerical experiments.

Contribution

It develops an unconditionally stable space-time isogeometric approach for biharmonic wave problems with a stabilized formulation and efficient solver, extending space-time methods beyond second order wave equations.

Findings

The method is stable under a CFL condition.

The stabilized formulation achieves unconditional stability.

Numerical experiments confirm convergence and efficiency.

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.