##
**A constructive a priori error estimation for finite element discretizations in a non-convex domain using singular functions.**
*(English)*
Zbl 1186.65143

This contribution provides constructively a priori error estimates for the finite element solution of elliptic problems, in particular Poisson’s equation, in domains which are bounded by orthogonal line segments and additionally have re-entrant corners. The rate of convergence of the solution of such problems is improved by adding singular functions to the usual interpolating basis. Explicit a priori \(H_0^1\) and \(L^2\) error estimates of \(O(h)\) and \(O(h^2)\), respectively, are derived for the finite element solution of such problems with \(\pi/2\) or \(3\pi/2\) corners, whereas all constants required are determined numerically. Two examples (L- and H-shaped domains) to confirm the validity of the error estimates conclude this clearly structured and well-written paper.

Reviewer: Gunnar Possart (Erlangen)

### MSC:

65N15 | Error bounds for boundary value problems involving PDEs |

65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |

35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |

65N50 | Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs |

65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |

### Keywords:

finite element method; a priori error estimation; Poisson equation; singular functions; numerical examples; re-entrant corners; convergence; L-shaped domains; H-shaped domains
PDF
BibTeX
XML
Cite

\textit{K. Kobayashi}, Japan J. Ind. Appl. Math. 26, No. 2--3, 493--516 (2009; Zbl 1186.65143)

### References:

[1] | R.A. Adams, Sobolev Spaces. Academic Press, New York, 1975. |

[2] | I. Babuska, R.B. Kellog and J. Pitkaranta, Direct and inverse error estimates for finite elements with mesh refinement. Numer. Math.,33 (1979), 447–471. · Zbl 0423.65057 |

[3] | H. Blum and M. Dobrowolski, On finite element methods for elliptic equations on domains with corners. Computing,28 (1982), 53–63. · Zbl 0461.65080 |

[4] | S.C. Brenner, Multigrid methods for the computation of singular solutions and stress intensity factors, I: Corner singularities. Math. Comp.,68 (1999), 559–583. · Zbl 1043.65136 |

[5] | Z. Cai and S. Kim, A finite element method using singular function for the Poisson equation: corner singularities. SIAM J. Numer. Anal.,39 (2001), 286–299. · Zbl 0992.65122 |

[6] | P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam, 1978. · Zbl 0383.65058 |

[7] | M. Dauge, S. Nicaise, M. Bourland and M.S. Lubuma, Coefficients of the singularities for elliptic boundary value problems on domains with conical points, III: Finite element methods on polygonal domains. SIAM J. Numer. Anal.,29 (1992), 136–155. · Zbl 0794.35015 |

[8] | G. Fix, S. Gulati and G.I. Wakoff, On the use of singular functions with the finite element method. J. Comp. Phys.,13 (1973), 209–228. · Zbl 0273.35004 |

[9] | P. Grisvard, Elliptic Problems in Nonsmooth Domains. Pitman Publishing, Boston, 1985. · Zbl 0695.35060 |

[10] | P. Grisvard, Singularities in Boundary Value Problems. RMA,22, Masson, Paris, 1992. · Zbl 0766.35001 |

[11] | O.A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flows. Gordon and Breach, 1969. · Zbl 0184.52603 |

[12] | M.T. Nakao, A numerical verification method for the existence of weak solutions for non-linear boundary value problems. J. Math. Anal. Appl.,164 (1992), 489–507. · Zbl 0771.65046 |

[13] | M. Schultz, Spline Analysis. Prentice-Hall, 1973. · Zbl 0333.41009 |

[14] | M. Tabata and M. Yamaguti, Approximate solution of the second order elliptic differential equation in a domain with piecewise smooth boundary by the finite-element method using singularity functions. Theor. and Appl. Mech.,22 (1974), 165–173. · Zbl 0329.35021 |

[15] | H. Takahasi and M. Mori, Double exponential formulas for numerical integration. Publ. Res. Inst. Math. Soc.,9 (1974), 721–741. · Zbl 0293.65011 |

[16] | N. Yamamoto and M.T. Nakao, Numerical verifications of solutions for elliptic equations in nonconvex polygonal domains. Numer. Math.,65 (1993), 503–521. · Zbl 0797.65081 |

[17] | Y. Watanabe and M.T. Nakao, Numerical verifications of solutions for nonlinear elliptic equations. Japan J. Indust. Appl. Math.,10 (1993), 165–178. · Zbl 0784.65082 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.