Unfitted Spectral Element Method for Elliptic Interface Problems

Core Concepts

Proposing an unfitted spectral element method for solving elliptic interface and eigenvalue problems.

Abstract

The paper introduces an unfitted spectral element method combining spectral accuracy with flexibility. It addresses interface problems with low regularity solutions. The method combines spectral element accuracy with Nitsche's flexibility. The paper establishes optimal convergence rates for elliptic interface and eigenvalue problems. It incorporates tailored ghost penalty terms for robustness. Various numerical methods for interface problems are discussed, including body-fitted mesh and unfitted methods. The cut finite element method (CutFEM) is highlighted for its robustness. The paper aims to solve elliptic interface problems and interface eigenvalue problems using a novel unfitted spectral element method. Nodal basis functions from Legendre-Gauss-Lobatto points are emphasized for enhanced robustness. The introduction of a ghost penalty stabilization term with tailored parameters is proposed. The paper is organized into sections focusing on equations, formulations, stability analysis, error estimates, and numerical integration methods.

Stats

We establish optimal hp convergence rates for both elliptic interface problems and interface eigenvalue problems.

Quotes

"The primary challenge for interface problems is the low regularity of the solution across the interface." "The key idea behind this approach involves employing two distinct sets of basis functions on the interface elements."

Key Insights Distilled From

Unfitted Spectral Element Method for interfacial models

by Nicolas Gonz... at arxiv.org 03-27-2024

https://arxiv.org/pdf/2309.17027.pdf

Deeper Inquiries

How does the proposed unfitted spectral element method compare to traditional finite element methods

The proposed unfitted spectral element method offers a unique approach compared to traditional finite element methods. While traditional finite element methods require conforming meshes that precisely align with the interface, the unfitted spectral element method allows for more flexibility by using non-conforming meshes. This flexibility is achieved through the incorporation of tailored ghost penalty terms, which enhance the method's robustness in handling geometrically complex interfaces. Additionally, the spectral accuracy of the spectral element method provides higher reliability in computing numerical eigenvalues, especially in scenarios where higher-order numerical methods are required for more accurate results.

What are the implications of the spectral accuracy demonstrated for model problems

The spectral accuracy demonstrated for model problems in the context of the unfitted spectral element method has significant implications for the reliability and efficiency of the numerical computations. By combining the spectral accuracy of the spectral element method with the flexibility of the unfitted Nitsche's method, the proposed approach can achieve higher convergence rates and more accurate results for both elliptic interface problems and interface eigenvalue problems. This spectral accuracy allows for more reliable numerical eigenvalues, especially in scenarios where higher-order numerical methods are necessary for precise computations.

How can the ghost penalty method enhance the robustness of the unfitted spectral element method

The ghost penalty method plays a crucial role in enhancing the robustness of the unfitted spectral element method. By incorporating the ghost penalty term with parameters tailored to the polynomial order, the method can effectively stabilize the solution, particularly in the presence of arbitrarily small intersections between geometric and numerical meshes. This stabilization helps mitigate issues that may arise from small cut geometries and ensures the accuracy of the error estimates derived from the method. Overall, the ghost penalty method contributes to the method's reliability and efficiency in solving interface problems.