[Paper Review] A superconvergent HDG method for the Incompressible Navier-Stokes Equations on general polyhedral meshes

Weifeng Qiu, Ke Shi|arXiv (Cornell University)|Jun 24, 2015

Advanced Numerical Methods in Computational Mathematics28 references3 citations

TL;DR

This paper presents a superconvergent hybridizable discontinuous Galerkin (HDG) method for the steady incompressible Navier-Stokes equations on general polyhedral meshes. Using $k+1$-degree polynomials for velocity, $k$-degree for velocity gradient and pressure, and $k$-degree for the numerical trace, the method achieves optimal $L^2$ convergence of order $k+1$ for all variables and superconvergence of order $k+2$ in the $L^2$-norm for velocity when $k \geq 1$, without postprocessing, under a small data condition.

ABSTRACT

We present a superconvergent hybridizable discontinuous Galerkin (HDG) method for the steady-state incompressible Navier-Stokes equations on general polyhedral meshes. For arbitrary conforming polyhedral mesh, we use polynomials of degree k+1, k, k to approximate the velocity, velocity gradient and pressure, respectively. In contrast, we only use polynomials of degree k to approximate the numerical trace of the velocity on the interfaces. Since the numerical trace of the velocity field is the only globally coupled unknown, this scheme allows a very efficient implementation of the method. For the stationary case, and under the usual smallness condition for the source term, we prove that the method is well defined and that the global L2-norm of the error in each of the above-mentioned variables and the discrete H1-norm of the error in the velocity converge with the order of k+1 for k>=0. We also show that for k>=1, the global L2-norm of the error in velocity converges with the order of k+2. From the point of view of degrees of freedom of the globally coupled unknown: numerical trace, this method achieves optimal convergence for all the above-mentioned variables in L2-norm for k>=0, superconvergence for the velocity in the discrete H1-norm without postprocessing for k>=0, and superconvergence for the velocity in L2-norm without postprocessing for k>=1.

Motivation & Objective

• To develop a high-order, efficient HDG method for the incompressible Navier-Stokes equations on general polyhedral meshes.
• To achieve superconvergence for velocity in both $L^2$ and discrete $H^1$ norms without postprocessing.
• To ensure optimal convergence rates for all variables—velocity, velocity gradient, pressure, and numerical trace—under minimal regularity assumptions.
• To extend the applicability of HDG methods to general polyhedral elements, including non-convex ones, while maintaining high-order accuracy.

Proposed method

• The method uses $k+1$-degree polynomials for velocity, $k$-degree for velocity gradient and pressure, and $k$-degree for the numerical trace of velocity on element interfaces.
• A modified numerical flux is introduced to enhance stability and convergence properties.
• The velocity gradient is reconstructed via a local element-wise solve using the velocity and numerical trace.
• The scheme is formulated via a hybridized weak form that couples unknowns only through the numerical trace on the mesh skeleton.
• Global coupling is minimized by solving for the numerical trace as the only globally coupled variable, enabling efficient solution procedures.
• The method employs a small data condition to ensure well-posedness and convergence under the given polynomial approximation.

Experimental results

Research questions

• RQ1 Can a superconvergent HDG method be constructed for the incompressible Navier-Stokes equations on general polyhedral meshes?
• RQ2 Does enriching the velocity approximation space to degree $k+1$ while using degree $k$ for other variables yield optimal and superconvergent convergence rates?
• RQ3 Can the method achieve superconvergence in the $L^2$-norm for velocity without postprocessing for $k \geq 1$?
• RQ4 Is the method well-posed and convergent under a smallness condition on the source term for general polyhedral meshes?

Key findings

• The method achieves optimal $L^2$-norm convergence of order $k+1$ for velocity, velocity gradient, pressure, and the numerical trace for all $k \geq 0$.
• For $k \geq 1$, the $L^2$-norm error in velocity converges with order $k+2$, demonstrating superconvergence without postprocessing.
• The discrete $H^1$-norm error in velocity converges with order $k+1$ for $k \geq 0$, indicating superconvergence without postprocessing.
• The method is well-defined and convergent under the standard smallness condition on the source term for arbitrary conforming polyhedral meshes.
• The analysis holds for shape-regular, possibly non-convex polyhedral elements, extending the applicability of HDG methods to general meshes.