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EFFICIENT PARAMETERS OF DECOUPLED DUAL SINGULAR FUNCTION METHOD
SEOKCHAN KIM,JAE-HONG PYO 한국산업응용수학회 2009 Journal of the Korean Society for Industrial and A Vol.13 No.4
The solution of the interface problem or Poisson problem with concave corner has singular perturbation at the interface corners or singular corners. The decoupled dual singular function method (DDSFM) which exploits the singular representations of the solutions was suggested in [3, 9] and estimated optimal accuracy in [10]. The convergence rates consist with theoretical results even for the problems with very strong singularity, with the efficiency depending on parameters used in the methods. Furthermore the errors in L² and L<SUP>∞</SUP>-spaces display some oscillation, in the cases with meshsize not small enough. In this paper, we present an answer to remove the oscillation via numerical experiments. We observe the effects of parameters in DDSFM, and show the consisting efficiency of the method over the strong singularity.
REMARKS ON FINITE ELEMENT METHODS FOR CORNER SINGULARITIES USING SIF
Kim, Seokchan,Kong, Soo Ryun The Honam Mathematical Society 2016 호남수학학술지 Vol.38 No.3
In [15] 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 we could get accurate solution just by adding the singular part. This approach works for the case when we have the accurate stress intensity factor. In this paper we consider Poisson equations with mixed boundary conditions and show the method depends the accrucy of the stress intensity factor by considering two algorithms.
FINITE ELEMENT SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATION WITH MULTIPLE CONCAVE CORNERS
( Seokchan Kim ),( Gyungsoo Woo ) 호남수학회 2018 호남수학학술지 Vol.40 No.4
In [8] they introduced a new _nite element method for accurate numerical solutions of Poisson equations with corner sin- gularities. They consider the Poisson equations with homogeneous Dirichlet boundary condition with one corner singularity at the ori- gin, and compute the _nite element solution using standard FEM and use the extraction formula to compute the stress intensity fac- tor, then pose a PDE with a regular solution by imposing the non- homogeneous boundary condition using the computed stress inten- sity factor, which converges with optimal speed. From the solution they could get an accurate solution just by adding the singular part. This approach uses the polar coordinate and the cut-off function to control the singularity and the boundary condition. In this paper we consider Poisson equations with multiple sin- gular points, which involves different cut-off functions which might overlaps together and shows the way of cording in FreeFEM++ to control the singular functions and cut-off functions with numerical experiments.
A FINITE ELEMENT METHOD USING SIF FOR CORNER SINGULARITIES WITH AN NEUMANN BOUNDARY CONDITION
Kim, Seokchan,Woo, Gyungsoo The Youngnam Mathematical Society 2017 East Asian mathematical journal Vol.33 No.1
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.
A FINITE ELEMENT METHOD USING SINGULAR FUNCTIONS FOR HELMHOLTZ EQUATIONS: PART I
SEOKCHAN KIM,JAE-HONG PYO,JONGSIK LEE 한국산업응용수학회 2008 Journal of the Korean Society for Industrial and A Vol.12 No.1
In [7, 8], they proposed a new singular function(NSF) method to compute singular solutions of Poisson equations on a polygonal domain with re-entrant angles. Singularities are eliminated and only the regular part of the solution that is in H² is computed. The stress intensity factor and the solution can be computed as a post processing step. This method was extended to the interface problem and Poisson equations with the mixed boundary condition. In this paper, we give NSF method for the Helmholtz equations -△u+Ku = f with homogeneous Dirichlet boundary condition. Examples with a singular point are given with numerical results.
REMARKS ON FINITE ELEMENT METHODS FOR CORNER SINGULARITIES USING SIF
( Seokchan Kim ),( Soo Ryun Kong ) 호남수학회 2016 호남수학학술지 Vol.38 No.3
In [15] 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 we could get accurate solution just by adding the singular part. This approach works for the case when we have the accurate stress intensity factor. In this paper we consider Poisson equations with mixed boundary conditions and show the method depends the accrucy of the stress intensity factor by considering two algorithms.
FINITE ELEMENT SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATION WITH MULTIPLE CONCAVE CORNERS
Kim, Seokchan,Woo, Gyungsoo The Honam Mathematical Society 2018 호남수학학술지 Vol.40 No.4
In [8] they introduced a new finite element method for accurate numerical solutions of Poisson equations with corner singularities. They consider the Poisson equations with homogeneous Dirichlet boundary condition with one corner singularity at the origin, and compute the finite element solution using standard FEM and use the extraction formula to compute the stress intensity factor, then 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 an accurate solution just by adding the singular part. This approach uses the polar coordinate and the cut-off function to control the singularity and the boundary condition. In this paper we consider Poisson equations with multiple singular points, which involves different cut-off functions which might overlaps together and shows the way of cording in FreeFEM++ to control the singular functions and cut-off functions with numerical experiments.