Convergence of adaptive BEM for some mixed boundary value problem

Area of Science:

• Computational Mathematics
• Numerical Analysis
• Partial Differential Equations

Background:

• The 2D Laplace equation is fundamental in various scientific fields.
• Mixed boundary conditions present unique challenges in numerical solutions.
• Boundary Element Methods (BEM) offer efficient solutions for certain boundary value problems.

Purpose of the Study:

• To develop and analyze an adaptive Galerkin BEM for the 2D Laplace equation with mixed boundary conditions.
• To incorporate Dirichlet, Neumann, and volume data resolution into the adaptive algorithm.
• To prove the convergence of the proposed adaptive scheme.

Main Methods:

• Adaptive Galerkin Boundary Element Method (BEM).
• Utilizing an [Formula: see text]-type error estimator for adaptivity.
• Implementation involving discrete integral operators only.
• Analysis of error estimators and solution convergence.

Main Results:

• The adaptive scheme generates a sequence of discrete solutions.
• Error estimators demonstrably tend to zero.
• Empirical observation of a saturation assumption for the non-perturbed problem.
• Convergence of discrete solutions to the exact solution in the energy norm is proven.

Conclusions:

• The proposed adaptive Galerkin BEM is effective for solving the 2D Laplace equation with mixed boundary conditions.
• The method guarantees convergence to the exact solution under empirically validated assumptions.
• The implementation's reliance on discrete integral operators simplifies practical application.