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.

翻译：缩放边界精度法( SBFEM) 是一种技术, 使用半分析法构建近距离空间。 它们基于多边形/多光度次区域的计算域分区, 形状函数与局部的 Dirichlet 问题相近, 与片断多感跟踪数据相近。 使用此操作器的调适方法, 并通过对各次区域强制一个类似星级的缩放要求, 本地 SBFEM 形状功能在辐射和表面方向中的表示来自一个 ODE 系统中的双向趋近值和偏差, 其系数由元素几何测距和微微多感空间的计算精确度确定。 本文的目的是为 SBFFEM 的调解析方法得出一个前置错误估计。 SBFFIEM 空间在 Duffy 的近似缩放标准中, 要求使用渐变多度调调调调调度制约。 因此, 缩放的边界函数是平流度和任何在 Duffy 空间中, 消失的数值由元素的精度精确度和微度 软度 软度 软度 软度 软度 度 测试中, 将SEMmlal 的精度的精度 的精度 的精度判判判判判判判判中, 的精度的精度的精度的精度的精度判判判法值校正度判算法, 。