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A Local Projection Stabilised HHO Method for the Oseen Problem

Gouranga Mallik, Rahul Biswas, Thirupathi Gudi

Abstract

In this article, we consider a local projection stabilisation for a Hybrid High-Order (HHO) approximation of the Oseen problem. We prove an existence-uniqueness result under a stronger SUPG-like norm. We improve the stability and provide error estimation in stronger norm for convection dominated Oseen problem. We also derive an optimal order error estimate under the SUPG-like norm for equal-order polynomial discretisation of velocity and pressure spaces. Numerical experiments are performed to validate the theoretical results.

A Local Projection Stabilised HHO Method for the Oseen Problem

Abstract

In this article, we consider a local projection stabilisation for a Hybrid High-Order (HHO) approximation of the Oseen problem. We prove an existence-uniqueness result under a stronger SUPG-like norm. We improve the stability and provide error estimation in stronger norm for convection dominated Oseen problem. We also derive an optimal order error estimate under the SUPG-like norm for equal-order polynomial discretisation of velocity and pressure spaces. Numerical experiments are performed to validate the theoretical results.
Paper Structure (10 sections, 6 theorems, 130 equations, 3 figures)

This paper contains 10 sections, 6 theorems, 130 equations, 3 figures.

Table of Contents

  1. Continuous Problem, Notations and Preliminaries
  2. Discrete Oseen Problem with LPS Stabilisation
  3. Local Reconstructions
  4. A Local Projection Setting
  5. Discrete Formulation
  6. Wellposedness of Discrete Formulation
  7. A Priori Error Estimates
  8. Numerical Results
  9. Conclusions
  10. Appendix

Key Result

Lemma 3.1

For given $(\ul_h,p_h) \in \underline{\boldsymbol{U}}_{h,0}^k \times P_h^k$, the bilinear form defined in defn_bilin satisfies

Figures (3)

  • Figure 1: (a) Triangular, (b) Cartesian and (c) hexagonal initial meshes.
  • Figure 2: Convergence histories for the error (left part - (a), (c) and (e)) in $\left| \! \left| \! \left| \cdot\right| \! \right| \! \right|_{\rm LP}$ norm of Example \ref{['test:smooth']} on the triangular, Cartesian, and hexagonal meshes for $\epsilon=10^{-8}$ and in the supg norm (right part - (b), (d) and (f)) $\|\cdot\|_{\rm supg}$ for $k=0,1,2,3$.
  • Figure 3: Convergence histories for the error (left part - (a), (c) and (e)) in $\left| \! \left| \! \left| \cdot\right| \! \right| \! \right|_{\rm LP}$ norm of Example \ref{['test:exp_layer']} on the triangular, Cartesian, and hexagonal meshes for $\epsilon=10^{-2}$ and in the supg norm (right part - (b), (d) and (f)) $\|\cdot\|_{\rm supg}$ for $k=1,2,3$.

Theorems & Definitions (17)

  • Remark 2.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Theorem 3.4
  • proof
  • Theorem 4.1
  • ...and 7 more