A high order accurate and energy stable continuous Galerkin framework on summation-by-parts form for the incompressible Navier-Stokes equations

TL;DR

This paper introduces a high-order, energy-stable continuous Galerkin finite element method for the incompressible Navier-Stokes equations, using SBP form and SAT boundary conditions, validated through numerical tests.

Contribution

It develops a novel SBP-SAT formulation for CGFEM that ensures energy stability and high accuracy for solving incompressible Navier-Stokes problems.

Findings

Achieved 4th order convergence with MMS.

Demonstrated accurate, oscillation-free solutions for discontinuous boundary conditions.

Validated efficiency on classical benchmark problems.

Abstract

This paper presents a high-order accurate Continuous Galerkin Finite Element Method (CGFEM) for solving the initial boundary value problems governed by the Incompressible Navier-Stokes (INS) equations. We discretize the INS equations using the CGFEM approach in Summation-By-Parts (SBP) form. Lagrange polynomials of up to 4th order are employed. The boundary conditions are imposed weakly using the Simultaneous Approximation Term (SAT) technique, which accommodates discontinuous boundary data without special treatment. The resulting SBP-SAT formulation guarantees an energy stable discretization. The efficiency of the proposed framework is demonstrated by solving a series of numerical tests. Initially, the Method of Manufactured Solutions (MMS) is employed to demonstrate 4th order convergence. Subsequently, the 4th order accurate scheme is applied to a classical benchmark problem featuring discontinuous boundary conditions: the lid-driven cavity flow over a wide range of Reynolds numbers. Accurate and oscillation-free solutions are achieved even in the vicinity of the discontinuous top corner boundaries. Lastly, a canonical backward-facing step flow problem is solved, where accuracy and efficiency are demonstrated.