Error estimates for the Scaled Boundary Finite Element Method

The Scaled Boundary Finite Element Method (SBFEM) is a technique in which approximation spaces are constructed using semi-analytical approach. They based on partitions of the computational domain by polygonal/polyhedral subregions, where shape functions approximate local Dirichlet problems with piecewise polynomial trace data. Using this operator adaptation approach, and imposing starlike scaling requirement representation SBFEM radial surface directions obtained from eigenvalues eigenfunctions an ODE system, whose coefficients determined element geometry spaces. aim paper to derive priori error estimates for SBFEM’s solutions harmonic test problems. For that, characterized context Duffy’s approximations gradient-orthogonality constraint imposed. As consequence, scaled boundary gradient-orthogonal any function vanishing at mesh skeleton, mimetic version well-known property valid functions. This orthogonality applied provide terms known finite interpolant errors exact solution. Similarities virtual also explored understanding convergence properties. Numerical experiments 2D 3D polytopal meshes confirm optimal rates two smooth solutions. Attention paid point singular solution close singularity elsewhere, revealing accuracy standard regular contexts.