Convergence analysis of the scaled boundary finite element method for the Laplace equation

The scaled boundary finite element method (SBFEM) is a relatively recent boundary element method that allows the approximation of solutions to partial differential equations (PDEs) without the need of a fundamental solution. A theoretical framework for the convergence analysis of SBFEM is proposed here. This is achieved by defining a space of semi-discrete functions and constructing an interpolation operator onto this space. We prove error estimates for this interpolation operator and show that optimal convergence to the solution can be obtained in SBFEM. These theoretical results are backed by two numerical examples.

Automated summary

SBFEM is a boundary element method for approximating solutions to PDEs without needing a fundamental solution. A theoretical framework for convergence analysis of SBFEM is proposed, using a space of semi-discrete functions and an interpolation operator. Error estimates for this operator are proved, demonstrating optimal convergence in SBFEM, supported by two numerical examples.