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L2-NORM ERROR ANALYSIS OF THE HP-VERSION WITH NUMERICAL INTEGRATION

  • Kim, Ik-Sung (Department of Applied Mathematics, Korea Maritime University)
  • Published : 2002.02.01

Abstract

We consider the hp-version to solve non-constant coefficient elliptic equations with Dirichlet boundary conditions on a bounded, convex polygonal domain $\Omega$ in $R^{2}.$ To compute the integrals in the variational formulation of the discrete problem we need the numerical quadrature rule scheme. In this paler we consider a family $G_{p}= {I_{m}}$ of numerical quadrature rules satisfying certain properties. When the numerical quadrature rules $I_{m}{\in}G_{p}$ are used for calculating the integrals in the stiffness matrix of the variational form we will give its variational fore and derive an error estimate of ${\parallel}u-\tilde{u}^h_p{\parallel}_0,{\Omega}'$.

References

  1. RAIRO Math. Mod. and Num. Anal. v.21 The hp-version of the finite element method with quasiuniform meshes https://doi.org/10.1051/m2an/1987210201991
  2. The finite·element method for elliptic problems P. G. Ciarlet
  3. Math. Mod. and Numer. Anal. v.24 The p-version of the finite element method for elliptic equations of order 21 M. Suri https://doi.org/10.1051/m2an/1990240202651
  4. Numer. Math. v.37 Error estimates for the combined h and p version of the finite element method I. Babuska;M. R. Dorr
  5. J. Appl. Numer. Math. v.6 Approximation results for spectral methods with domain decomposition C. Bernardi;Y. Maday https://doi.org/10.1016/0168-9274(89)90053-6
  6. SIAM J. Numer. Anal. v.24 The optimal convergence rate of the p-version of the finite element method https://doi.org/10.1137/0724049
  7. Sobolev spaces R. A. Adams
  8. Math. of comput. v.38 Approximation Results for orthogonal polynomials in sobolev spaces C. Canuto;A. Quarteroni https://doi.org/10.1090/S0025-5718-1982-0637287-3
  9. Technical Note BN-1128, Institute for Phy. Sci. and Tech. Performance of the h-p version of the finite element method with various elements I. Babuska;H. C. Elman
  10. Technical Note BN-1101, Institute for Phy. Sci. and Tech. The p and h-p versions of the finite element method I. Babuska;M. Suri
  11. An introduction Interpolation spaces J. Bergh;J. Lofstrom