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arXiv 2410.16596math.NAcs.NA
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基于区间上的双正交小波的伽辽金方案用于椭圆界面问题

Galerkin Scheme Using Biorthogonal Wavelets on Intervals for Elliptic Interface Problems

University of Alberta（阿尔伯塔大学）

Bin Han, Michelle Michelle

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AI总结：

本文提出一种小波伽辽金方法，用于求解椭圆界面问题，通过在界面处加入额外的小波基函数来处理系数和源项的间断性，实现近最优收敛率。

摘要：

本文提出了一种小波伽辽金方法，用于求解形式为$-\nabla\cdot(a\nabla u)=f$在$Ω\backslash Γ$中的椭圆界面问题，其中$Γ$是$Ω$内的光滑界面。由于标量系数$a>0$和源项$f$在$Γ$处通常间断，解$u$的梯度$\nabla u$通常在$Γ$处间断，因此$u\not\in H^{1.5}(Ω)$，这对传统数值方法构成重大挑战。通过利用用于$H^1_0(Ω)$的紧支撑双正交小波，我们开发了一种策略，通过在界面处加入额外的小波元素（或基函数）来解决界面$Γ$的复杂几何和由此产生的梯度间断。对于二维椭圆界面问题，所提出的方法在$H^1(Ω)$范数中达到近最优收敛率：$\mathcal{O}(h |\log(h)|)$，在$L^{2}$范数中为$\mathcal{O}(h^2 |\log(h)|^2)$，与近似阶数相关。关键的理论贡献是使用双正交小波基来建立$H^1(Ω)$收敛结果。这通过开发加权Bessel性质和几个分数Sobolev空间中的不等式得到支持。为了保持高精度和对高对比系数的鲁棒性，我们的方法利用了增强的小波元素集，类似于无网格方法，从而消除了有限元方法所需的复杂重新网格化需求。与现有技术不同，这种小波Riesz基框架能够无缝捕捉$Γ$的几何，同时确保系数矩阵的条件数保持小且统一有界，与问题规模无关。

英文摘要：

This paper presents a wavelet Galerkin method for solving elliptic interface problems of the form $-\nabla\cdot(a\nabla u)=f$ in $Ω\backslash Γ$, where $Γ$ is a smooth interface within $Ω$. Since the scalar variable coefficient $a>0$ and source term $f$ are often discontinuous across $Γ$, the solution $u$ typically has discontinuous gradient $\nabla u$ across $Γ$ and hence $u\not\in H^{1.5}(Ω)$, posing significant challenges for traditional numerical methods. By utilizing a compactly supported biorthogonal wavelet for $H^1_0(Ω)$, we develop a strategy that incorporates additional wavelet elements (or basis functions) along the interface to resolve the complex geometry of the interface $Γ$ and the resulting gradient discontinuities. For the two-dimensional (2D) elliptic interface problem, the proposed method achieves near-optimal convergence rates: $\mathcal{O}(h |\log(h)|)$ in the $H^1(Ω)$-norm and $\mathcal(h^2 |\log(h)|^2)$ in the $L^{2}$-norm with respect to the approximation order. A key theoretical contribution is the use of the dual biorthogonal wavelet basis to establish the $H^1(Ω)$ convergence results. This is supported by the development of weighted Bessel properties for wavelets and several inequalities in fractional Sobolev spaces. To maintain high accuracy and robustness against high-contrast coefficients, our method leverages an augmented set of wavelet elements, similar to meshfree approaches, thereby eliminating the need for the complex re-meshing required by finite element methods. Unlike existing techniques, this wavelet Riesz basis framework captures the geometry of $Γ$ seamlessly while ensuring that the condition numbers of the coefficient matrices remain small and uniformly bounded, independent of the problem size.

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