A consistent and stabilized continuous/discontinuous Galerkin method for fourth-order incompressible flow problems. (English) Zbl 1457.65103
Summary: This paper presents a new consistent and stabilized finite-element formulation for fourth-order incompressible flow problems. The formulation is based on the \(C^{0}\)-interior penalty method, the Galerkin least-square (GLS) scheme, which assures that the formulation is weakly coercive for spaces that fail to satisfy the inf-sup condition, and considers discontinuous pressure interpolations. A stability analysis through a lemma establishes that the proposed formulation satisfies the inf-sup condition, thus confirming the robustness of the method. This lemma indicates that, at the element level, there exists an optimal or quasi-optimal GLS stability parameter that depends on the polynomial degree used to interpolate the velocity and pressure fields, the geometry of the finite element, and the fluid viscosity term. Numerical experiments are carried out to illustrate the ability of the formulation to deal with arbitrary interpolations for velocity and pressure, and to stabilize large pressure gradients.
MSC:
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 76D05 | Navier-Stokes equations for incompressible viscous fluids |
| 76M10 | Finite element methods applied to problems in fluid mechanics |
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