2024年07月17日 星期三 登录 EN

学术活动
Spline Wavelet Methods for Helmholtz Equations and Elliptic Interface Problems
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报告人:
Bin Han, Professor, University of Alberta, Canada
邀请人:
Zhiqiang Xu, Professor
题目:
Spline Wavelet Methods for Helmholtz Equations and Elliptic Interface Problems
时间地点:
15:00-16:00 July 11(Thursday), N933
摘要:

Wavelets have been applied to numerical PDEs since 1990's and offer the advantage of achieving uniformly bounded condition numbers in various Sobolev spaces. However, the performance of wavelet methods for PDEs is only comparable with finite element methods coupled with the multigrid and multilevel methods. In this talk, we briefly discuss how to construct wavelets on the interval $[0,1]$ (with or without homogeneous boundary conditions) and then their tensor products offer Riesz bases in the Sobolev space $H^1(\Omega)$ with $\Omega=(0,1)^d$. Next we apply wavelets to the large cavity problem modeled by the Helmholtz equation and the elliptic interface problems. Avoiding complicated handling of the interface, we apply wavelets to the elliptic interface problems by a simple strategy (not adapted to the interface). In dimension one, we can prove that the wavelet method can achieve convergence rate $O(h^m)$ in $H^1(\Omega)$ for arbitrary $m$. In dimension $d\ge 2$, the wavelet method can achieve the convergence rate $O(h^{\frac{0.5 d}{d-1}})$ in $H^1(\Omega)$. In 2D, the scheme is order 1 in $H^1$ norm and order 2 in $L_2$ norm). As a meshfree method, the condition numbers of wavelet methods for both Helmholtz equations and elliptic interface problems are uniformly bounded. Some initial numerical experiments are provided to illustrate the performance of wavelet methods for both the Helmholtz equation and the elliptic interface problem. For comparison purposes, we shall also solve these two problems using high-order finite difference methods (FDMs) and explain the advantages/disadvantages of wavelet methods over FDMs. This talk is built on several ongoing research projects with Dr. M. Michelle at Purdue University.