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hp Discontinuous Galerkin Approximations for the Stokes Problem

by A. Toselli

(Report number 2002-02)

Abstract
We propose and analyze a discontinuous Galerkin approximation for the Stokes problem. The finite element triangulation employed is not required to be conforming and we use discontinuous pressures and velocities. No additional unknown fields need to be introduced, but only suitable bilinear forms defined on the interfaces between the elements, involving the jumps of the velocity and the average of the pressure. We consider hp approximations using Qk'-Qk velocity-pressure pairs with k'=k+2, k+1, k. Our methods show better stability properties than the corresponding conforming ones. We prove that our first two choices of velocity spaces ensure uniform divergence stability with the respect to the mesh size h. Numerical results show that they are uniformly stable with respect to the local polynomial degree k, a property that has no analog in the conforming case. An explicit bound in k which is not sharp is also proven. Numerical results show that if equal order approximation is chosen for the velocity and pressure, no spurious pressure modes are present but the method is not uniformly stable either with respect to h or k. We derive a priori error estimates generalizing the abstract theory of mixed methods. Optimal error estimates in h are proven. As for discontinuous Galerkin methods for scalar diffusive problems, half power of k is lost for p and hp approximations independently of the divergence stability.

Keywords: Mixed problems, hp approximations, spectral elements, discontinuous Galerkin approximations, non-conforming approximations

@Techreport{T02_288,
  author = {A. Toselli},
  title = {hp Discontinuous Galerkin Approximations for the Stokes Problem},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2002-02},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2002/2002-02.pdf },
  year = {2002}
}

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