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Yuan, Si; Wu, Yue; Xing, Qinyan

Recursive super-convergence computation for multi-dimensional problems via one-dimensional element energy projection technique. (English) Zbl 1397.65122

AMM, Appl. Math. Mech., Engl. Ed. 39, No. 7, 1031-1044 (2018).
Summary: This paper presents a strategy for computation of super-convergent solutions of multi-dimensional problems in the finite element method (FEM) by recursive application of the one-dimensional (1D) element energy projection (EEP) technique. The main idea is to conceptually treat multi-dimensional problems as generalized 1D problems, based on which the concepts of generalized 1D FEM and its consequent EEP formulae have been developed in a unified manner. Equipped with these concepts, multi-dimensional problems can be recursively discretized in one dimension at each step, until a fully discretized standard finite element (FE) model is reached. This conceptual dimension-by-dimension (D-by-D) discretization procedure is entirely equivalent to a full FE discretization. As a reverse D-by-D recovery procedure, by using the unified EEP formulae together with proper extraction of the generalized nodal solutions, super-convergent displacements and first derivatives for two-dimensional (2D) and three-dimensional (3D) problems can be obtained over the domain. Numerical examples of 3D Poisson’s equation and elasticity problem are given to verify the feasibility and effectiveness of the proposed strategy.

MSC:

65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
65L50 Mesh generation, refinement, and adaptive methods for ordinary differential equations
65L10 Numerical solution of boundary value problems involving ordinary differential equations
74S05 Finite element methods applied to problems in solid mechanics

Keywords:

three-dimensional (3D) problem; generalized one-dimensional (1D) finite element method (FEM); dimension-by-dimension (D-by-D); super-convergence; element energy projection (EEP)
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References:

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