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Singular solutions of the Poisson equation at edges of three‐dimensional domains and their treatment with a predictor–corrector finite element method
Author(s) -
Nkemzi Boniface,
Jung Michael
Publication year - 2021
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.22555
Subject(s) - mathematics , finite element method , gravitational singularity , poisson's equation , rate of convergence , superconvergence , convergence (economics) , mathematical analysis , a priori and a posteriori , boundary value problem , mixed finite element method , boundary (topology) , polygon mesh , geometry , computer science , key (lock) , philosophy , physics , computer security , epistemology , economics , thermodynamics , economic growth
Solutions of boundary value problems in three‐dimensional domains with edges may exhibit singularities which are known to influence both the accuracy of the finite element solutions and the rate of convergence in the error estimates. This paper considers boundary value problems for the Poisson equation on typical domains Ω ⊂ ℝ 3 with edge singularities and presents, on the one hand, explicit computational formulas for the flux intensity functions. On the other hand, it proposes and analyzes a nonconforming finite element method on regular meshes for the efficient treatment of the singularities. The novelty of the present method is the use of the explicit formulas for the flux intensity functions in defining a postprocessing procedure in the finite element approximation of the solution. A priori error estimates in H 1 (Ω) show that the present algorithm exhibits the same rate of convergence as it is known for problems with regular solutions.