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