A finite element method for computing accurate solutions for Poisson equations with corner singularities using the stress intensity factor

Kim, Seokchan; Lee, Hyung-Chun

doi:10.1016/j.camwa.2015.12.023

Cited by 12 publications

(21 citation statements)

References 15 publications

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“…We give three results of numerical experiments, including the standard finite element method, the dual singular function method using the equation (5), and the one similar to that in [8]. An example will be given in Section 4 with computational results using FreeFEM++ code ( [6]).…”

Section: Introductionmentioning

confidence: 99%

“…Unfortunately, the results were not good enough because the input function f was replaced by f + λ∆(ηs), etc., whose L 2 − norms are quite large compared to that of f (see Lemma 2.2 in [8] ).…”

Section: Introductionmentioning

confidence: 99%

“…As observed in [8], some numerical approaches (e.g. [1,2,3]) use this extraction formula for λ and seek the regular part w ∈ H 2 (Ω) from a new partial differential equation, for example,…”

Section: Introductionmentioning

confidence: 99%

“…In this paper we consider a Poisson problem with a concave corner with the Neumann boundary condition and suggest a proper algorithm similar to that in [8] together with some theorems. We give three results of numerical experiments, including the standard finite element method, the dual singular function method using the equation (5), and the one similar to that in [8].…”

Section: Introductionmentioning

confidence: 99%

“…In [8] they introduced a new partial differential equation, whose solution is in H 2 (Ω) with the same input function by simple changing of the boundary condition. Using this partial differential equation, they suggested an efficient algorithm to compute the numerical solution for Poisson equation with Dirichlet boundary condition containing domain singularity.…”

Section: Introductionmentioning

confidence: 99%

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A Finite Element Method Using Sif for Corner Singularities With an Neumann Boundary Condition

Kim¹,

Woo²

2017

East Asian mathematical journal

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Abstract. In [8] they introduced a new finite element method for accurate numerical solutions of Poisson equations with corner singularities, which is useful for the problem with known stress intensity factor. They consider the Poisson equations with homogeneous Dirichlet boundary condition, compute the finite element solution using standard FEM and use the extraction formula to compute the stress intensity factor, then they pose a PDE with a regular solution by imposing the nonhomogeneous boundary condition using the computed stress intensity factor, which converges with optimal speed. From the solution they could get accurate solution just by adding the singular part. This approach works for the case when we have the reasonably accurate stress intensity factor. In this paper we consider Poisson equations defined on a domain with a concave corner with Neumann boundary conditions. First we compute the stress intensity factor using the extraction formular, then find the regular part of the solution and the solution.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A Finite Element Method Using Sif for Corner Singularities With an Neumann Boundary Condition

Kim¹,

Woo²

2017

East Asian mathematical journal

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Error estimate of a finite element method for an optimal control problem with corner singularity using the stress intensity factor

Kim

Lee

2021

Numerical Methods Partial

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We consider an optimal control problem for the Poisson equation on a non-convex polygonal domain with the corner singularity. Previously, we proposed a novel algorithm for the accurate numerical solution for the Poisson equation on a polygonal domain with the domain singularity. Then, we investigated the error estimate and its efficient procedure for the numerical algorithm. In this article, we propose an efficient algorithm and perform an error estimate for a distributed optimal control problem of the Poisson equation. The solutions of the optimality system with such singularity have singular decompositions: regular part plus singular part for each state variable and adjoint variable. The coefficient of the singular function is usually called stress intensity factor and can be computed by the extraction formula. We introduced a modified optimality system which has "zero" stress intensity factors using this stress intensity factor, from whose solutions we can compute very accurate solution of the original optimality system simply by adding a singular part. We give a precise error analysis and provide numerical results which justify the results therein.

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Construction of equilibrated singular basis functions without a priori knowledge of analytical singularity order

Noormohammadi

Boroomand

2017

Computers & Mathematics with Applications

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A finite element method for computing accurate solutions for Poisson equations with corner singularities using the stress intensity factor

Cited by 12 publications

References 15 publications

A Finite Element Method Using Sif for Corner Singularities With an Neumann Boundary Condition

A Finite Element Method Using Sif for Corner Singularities With an Neumann Boundary Condition

Error estimate of a finite element method for an optimal control problem with corner singularity using the stress intensity factor

Construction of equilibrated singular basis functions without a priori knowledge of analytical singularity order

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