Unfitted Spectral Element Method for interfacial models

Abstract

In this paper, we propose the unfitted spectral element method for solving elliptic interface and corresponding eigenvalue problems. The novelty of the proposed method lies in its combination of the spectral accuracy of the spectral element method and the flexibility of the unfitted Nitsche's method. We also use tailored ghost penalty terms to enhance its robustness. We establish optimal $hp$ convergence rates for both elliptic interface problems and interface eigenvalue problems. Additionally, we demonstrate spectral accuracy for model problems in terms of polynomial degree.

Figures (8)

• Figure 1: Left: $\Omega$ and interface $\Gamma$ (white curve described as a level-set). Green elements: $\Omega_{\Gamma,h}$. Blue elements: $\Omega_{-,h} \backslash \mathcal{T}_{\Gamma,h}$. Red elements: $\Omega_{+,h} \backslash \mathcal{T}_{\Gamma,h}$. Middle: Blue elements: $\Omega_{-,h}$. Yellow edges: $\mathcal{G}_{-,h}$. Right: Red elements: $\Omega_{+,h}$. Yellow edges: $\mathcal{G}_{+,h}$. Note that a ghost penalty face may belong to both $\Omega_-,\Omega_+$.
• Figure 1: Log-log $h$-convergence plots for circle interface problem with $\alpha_- = 1$ and $\alpha_+ = 1000$. Orange line: non-stabilized solution. Yellow Line: stabilized solution. (a) $L^2$-error; (b) $H^1$-error; (c) Condition number. Stabilized with $\gamma_A = 0.1$.
• Figure 2: Quadrature on interface element $K = [\tilde{x}_m,\tilde{x}_M] \times [\tilde{y}_m,\tilde{y}_M]$. $r_1,r_2$ are roots for functions $\tilde{f}_m, \tilde{f}_M$ respectively. Volume quadratures: blue nodes $(\bar{\bm{r}} \; \vert \; \check{\bm{\rho}})$ for $\Omega_-$ and red nodes $(\bar{\bm{r}} \; \vert \; \check{\bm{\rho}})$ for $\Omega_+$. Surface quadrature on $\Gamma$: yellow nodes, $\bm{\rho}$, also $f_j^*$'s root for each $j$.
• Figure 2: Semi-log $p$-convergence plots for circle interface problem with $\alpha_- = 1$ and $\alpha_+ = 1000$. Orange line: non-stabilized solution. Yellow Line: stabilized solution. (a) $L^2$-error; (b) $H^1$-error; (c) Condition number. Stabilized with $\gamma_A = 0.05$.
• Figure 3: Log-log $h$-convergence plots for flower shape interface problem with $\alpha_- = 1$ and $\alpha_+ = 10$. Orange line: non-stabilized solution. Yellow Line: stabilized solution. (a) $L^2$-error; (b) $H^1$-error; (c) Condition number. Stabilized with $\gamma_A = 0.001$.

Theorems & Definitions (24)

• Remark 3.2
• Remark 3.3
• Remark 3.4
• Theorem 3.5
• Proof 1
• Remark 3.6
• Remark 3.7
• Remark 3.8
• Theorem 4.1
• Proof 2