FIGURE 1. A domain with multiple concave corners and corresponding polar coordinates
FIGURE 2. Two polar coordinates on a T-shaped domain
TABLE 1. Errors and convergence rates of the λ1,h and λ2,h
TABLE 2. Errors and convergence rates for uh with the Standard FEM
TABLE 3. Errors and convergence rates for uh with our algorithmA
References
- I. BABUSKA, R.B. KELLOGG, AND J. PITKARANTA, Direct and inverse error estimates for finite elements with mesh refinements, Numer. Math., 33 (1979), 447-471. https://doi.org/10.1007/BF01399326
- H. BLUM AND M. DOBROWOLSKI, On finite element methods for elliptic equations on domains with corners, Computing, 28 (1982), 53-63. https://doi.org/10.1007/BF02237995
- Z. CAI AND S.C. KIM, A finite element method using singular functions for the poisson equation: Corner singularities, SIAM J. Numer. Anal., 39:(2001), 286-299. https://doi.org/10.1137/S0036142999355945
- Z. CAI, S.C. KIM, S.D. KIM, S. KONG, A finite element method using singular functions for Poisson equations: Mixed boundary conditions, Comput. Methods Appl. Mech. Engrg. 195 (2006) 26352648
- G. J. FIX, S. GULATI, AND G. I. WAKOFF, On the use of singular functions with finite elements approximations, J. Comput. Phy., 13 (1973), 209-228. https://doi.org/10.1016/0021-9991(73)90023-5
- P. GRISVARD, Elliptic Problems in Nonsmooth Domains, Pitman, Boston, MA, 1985.
- F. HECHT, New development in FreeFem++, J. Numer. Math. 20 (2012), no. 3-4, 251265.
- S. KIM AND H.-C. LEE, A finite element method for computing accurate solutions for Poisson equations with corner singularities using the stress intensity factor, Computers and Mathematics with Applications, 71(2016) 2330-2337. https://doi.org/10.1016/j.camwa.2015.12.023
- S. KIM AND H.-C. LEE, Finite element method to control the domain singularities of Poisson equation using the stress intensity factor : mixed boundary condition, Int. J. Numer. Anal. Model, 14:4-5 (2017), 500-510.