Galerkin Scheme Using Biorthogonal Wavelets on Intervals for 2D Elliptic Interface Problems

TL;DR

The paper develops a wavelet Galerkin scheme for 2D elliptic interface problems with discontinuous coefficients across a smooth interface Γ. It constructs a tensor-product biorthogonal wavelet basis on a unit square from bilinear elements, forming a Riesz basis for $H^1_0(\Omega)$, and augments the basis with interface-touching wavelets to capture Γ and the gradient singularities of the solution. The authors prove near-optimal convergence: $\|u-u_h\|_{L^2(\Omega)} \lesssim h^2|\log h|^2$ and $\|u-u_h\|_{H^1(\Omega)} \lesssim h|\log h|$, with discretization matrices having uniformly bounded condition numbers independent of mesh size and interface geometry. The method behaves in a meshfree-like manner, refining by increasing wavelet scale and adding targeted elements near the interface rather than remeshing. Numerical experiments validate the convergence rates and conditioning, including high-contrast diffusion and complex interface shapes, and demonstrate efficiency gains in GMRES iterations compared to standard FEM.

Abstract

This paper introduces a wavelet Galerkin method for solving two-dimensional elliptic interface problems of the form $-\nabla\cdot(a\nabla u)=f$ in $Ω\backslash Γ$, where $Γ$ is a smooth interface within $Ω$. The variable scalar coefficient $a>0$ and source term $f$ may exhibit discontinuities across $Γ$. By utilizing a biorthogonal wavelet basis derived from bilinear finite elements, which serves as a Riesz basis for $H^1_0(Ω)$, we devise a strategy that achieves nearly optimal convergence rates: $O(h^2 |\log(h)|^2)$ in the $L_2(Ω)$-norm and $O(h |\log(h)|)$ in the $H^1(Ω)$-norm with respect to the approximation order. To handle the geometry of $Γ$ and the singularities of the solution $u$, which has a discontinuous gradient across $Γ$, additional wavelet elements are introduced along the interface. The dual part of the biorthogonal wavelet basis plays a crucial role in proving these convergence rates. We develop weighted Bessel properties for wavelets, derive various inequalities in fractional Sobolev spaces, and employ finite element arguments to establish the theoretical convergence results. To achieve higher accuracy and effectively handle high-contrast coefficients $a$, our method, much like meshfree approaches, relies on augmenting the number of wavelet elements throughout the domain and near the interface, eliminating the need for re-meshing as in finite element methods. Unlike all other methods for solving elliptic interface problems, the use of a wavelet Riesz basis for $H^1_0(Ω)$ ensures that the condition numbers of the coefficient matrices remain small and uniformly bounded, regardless of the matrix size.

Figures (9)

• Figure 1: Panels (a)-(b) depict generators of the 1D wavelet basis $\mathcal{B}^{1D}_{J_0}$ with $J_0=3$. Panels (c)-(h) depict generators of the 2D wavelet basis $\mathcal{B}^{2D}_{J_0}$ with $J_0 =3$.
• Figure 2: For simplicity, we assume that the interface curve, $\Gamma$, is a circle. Panel (a) depicts the overlapping supports of wavelets in $\mathcal{B}^{2D}_{3,4}$. Panels (b)-(d) depict the overlapping supports of extra wavelets added along the interface $\Gamma$, which make up the set $\cup_{j=4}^{6} [2^{-j} \mathcal{S}_j]$.
• Figure 3: \ref{['ex:circle']}. Left: the plot of $\Gamma$. Middle: the plot of the approximated solution at $J=7$, where $a_+ = 10^6$. Right: the plot of the error at $J=7$, where $a_+ = 10^6$.
• Figure 4: \ref{['ex:star:gonglili']}. Left: the plot of $\Gamma$. Middle: the plot of the approximated solution at $J=7$ with $a_+ = 10^2$. Right: the plot of the approximated solution at $J=7$ with $a_- = 10^{-2}$.
• Figure 5: \ref{['ex:flower5']}. Left: the plot of $\Gamma$. Middle: the plot of the approximated solution at $J=7$. Right: the plot of the error at $J=7$.

Theorems & Definitions (16)

• Theorem 2.1
• Theorem 2.2
• Example 3.1
• Example 3.2
• Example 3.3
• Example 3.4
• Example 3.5
• Example 3.6
• Example 3.7
• Theorem 4.1