Galerkin Scheme Using Biorthogonal Wavelets on Intervals for Elliptic Interface Problems
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866911672405327872 |
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| author | Han, Bin Michelle, Michelle |
| author_facet | Han, Bin Michelle, Michelle |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_16596 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Galerkin Scheme Using Biorthogonal Wavelets on Intervals for Elliptic Interface Problems Han, Bin Michelle, Michelle Numerical Analysis 65T60, 35J15, 42C40, 41A15 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. |
| title | Galerkin Scheme Using Biorthogonal Wavelets on Intervals for Elliptic Interface Problems |
| topic | Numerical Analysis 65T60, 35J15, 42C40, 41A15 |
| url | https://arxiv.org/abs/2410.16596 |