
Convergence analysis of the scaled boundary finite element method for the Laplace equation
The scaled boundary finite element method (SBFEM) is a relatively recent...
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Error estimates for fully discrete generalized FEMs with locally optimal spectral approximations
This paper is concerned with the error estimates of the fully discrete g...
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Poisson Vector Graphics (PVG) and Its ClosedForm Solver
This paper presents Poisson vector graphics, an extension of the popular...
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Equivalence of localand globalbest approximations, a simple stable local commuting projector, and optimal hp approximation estimates in H(div)
Given an arbitrary function in H(div), we show that the error attained b...
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Error estimates for semidiscrete finite element approximations for a moving boundary problem capturing the penetration of diffusants into rubber
We study a semidiscrete finite element approximation of weak solutions ...
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Novel design and analysis of generalized FE methods based on locally optimal spectral approximations
In this paper, the generalized finite element method (GFEM) for solving ...
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A C^1continuous TraceFiniteCellMethod for linear thin shell analysis on implicitly defined surfaces
A TraceFiniteCellMethod for the numerical analysis of thin shells is ...
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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 a semianalytical 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 gradientorthogonality constraint is imposed. As a consequence, the scaled boundary functions are gradientorthogonal to any function in Duffy's spaces vanishing at the mesh skeleton, a mimetic version of a wellknown 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.
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