Hybrid Dual Mimetic Spectral Element Method with a Novel Dual Grid

Abstract

The mimetic spectral element method (MSEM) is a structure-preserving discretization scheme based on the Galerkin Method, which strongly constrains the topology relations by discretizing and reconstructing variables in specific function spaces in order to preserve certain critical structures of the PDE in the numerical solution. In studying the 2D incompressible Navier-Stokes equations, the conservation law of mass, energy, vorticity, and enstrophy (or helicity for 3D cases) are expected to be preserved. According to the de Rham complex, the mimetic spectral element method uses differential forms rather than vector or scalar fields to present physical variables and discretize differential forms on specified function spaces. It has two significant advantages. Firstly, the topological relations between discretized variables depend only on the grid's topology structure, which means no numerical errors are introduced into the discretized conservation equations. Secondly, the variables are reconstructed with spectral functions, which can be of arbitrary high order.

Based on the MSEM, a more efficient hybrid dual mimetic spectral element method (hdMSEM) was proposed. In the hybrid mimetic spectral element method (hMSEM), a set of trace function spaces and trace variables are introduced at the interface between subdomains, applying a Lagrange multiplier to strongly couple the variables of bordered subdomains so that domain decomposition is feasible, and the solver can run in parallel efficiently. In addition, designing a proper set of dual grid and dual function spaces for trace variables can further increase the sparsity of the matrix, thus saving computational resources. However, a singularity problem arises in the structure-preserving simulation of incompressible flows with the primal hdMSEM when Lagrange multipliers are applied to couple variables of vorticity at the edge where more than two subdomains meet.

This thesis proposes the hybrid dual mimetic spectral element method with a novel dual grid, which can avoid the singularity and simultaneously keep the matrix of the discrete system symmetrical and the mathematical definition of the matrix equations rigorous. The basic idea is to introduce a dummy degree of freedom at the edge where singularity arises and design a curvilinear dual grid for trace variables to couple the degree of freedom of vorticity and the dummy degree of freedom to eliminate singularity. Besides, this thesis studies the implementations of several kinds of boundary conditions with the novel dual grid and the corresponding grid topology near boundaries. Then, we extend the hdMSEM with the novel dual grid to solve steady and unsteady 2D incompressible Navier-Stokes equations. In numerical experiments, the accuracy and structure-preserving capability are verified numerically with several benchmark cases.