MIB Galerkin method for elliptic interface problems.

Area of Science:

• Computational Mathematics
• Numerical Analysis
• Partial Differential Equations

Background:

• Elliptic interface problems involve discontinuous coefficients and singular sources, crucial for modeling real-world material interfaces.
• Existing high-order numerical schemes struggle with nonsmooth interfaces, geometric singularities, and low solution regularity.
• Tip-geometry effects in various fields are amplified by singularities combined with low solution regularity.

Purpose of the Study:

• To develop a high-order numerical scheme for two-dimensional (2D) elliptic partial differential equations (PDEs) with complex interfaces.
• To address challenges posed by geometric singularities and low solution regularity in interface problems.
• To introduce a novel Matched Interface and Boundary (MIB) Galerkin method.

Main Methods:

• Employs a Cartesian grid-based approach with triangular elements to bypass complex mesh generation.
• Introduces Matched Interface and Boundary (MIB) elements to ensure basis function continuity across interfaces.
• Constructs interpolation functions on MIB element spaces and enforces interface jump conditions to determine these functions.

Main Results:

• The Matched Interface and Boundary (MIB) Galerkin method demonstrates second-order convergence in both L∞ and L2 errors.
• Numerical experiments validate the method's performance across diverse interface geometries, singularities, and solution regularities.
• Achieves high accuracy, particularly for C1 or Lipschitz continuous interfaces and C2 continuous solutions.

Conclusions:

• The proposed MIB Galerkin finite element method effectively solves 2D elliptic PDEs with complex, nonsmooth interfaces and low solution regularity.
• The method provides a robust and accurate solution strategy for challenging interface problems.
• Confirms the second-order accuracy and broad applicability of the MIB Galerkin approach.