Dynamic term-by-term stabilized finite element formulation using orthogonal subgrid-scales for the incompressible Navier–Stokes problem

• Published in: Computer methods in applied mechanics and engineering Vol. 349; pp. 701 - 721
• Main Authors: Castillo, E., Codina, R.
• Format: Journal Article Publication
• Language: English
• Published: Amsterdam Elsevier B.V 01.06.2019; Elsevier BV
• Subjects: Anàlisi numèrica; Computational fluid dynamics; Dynamic stability; Dynamic subscales; Equacions de Navier-Stokes; Finite element method; Fluid flow; Formulations; Matemàtiques i estadística; Mathematical analysis; Multiscale analysis; Mètodes en elements finits; Mètodes numèrics; Navier-Stokes equations; Numerical analysis; Potential fields; Stability analysis; Stabilized finite element methods; Stabilized finite element methods Variational multiscale Dynamic subscales Term-by-term stabilization; Structural stability; Term-by-term stabilization; Time dependence; Variational multiscale; Àrees temàtiques de la UPC; Variational multiscale; Term-by-term stabilization; Dynamic subscales; Stabilized finite element methods
• ISSN: 0045-7825; 1879-2138
• DOI: 10.1016/j.cma.2019.02.041

In this paper, we propose and analyze the stability and the dissipative structure of a new dynamic term-by-term stabilized finite element formulation for the Navier–Stokes problem that can be viewed as a variational multiscale (VMS) method under some assumptions. The essential point of the formulation is the time dependent nature of the subscales and, contrary to residual-based formulations, the introduction of two velocity subscale components. They represent the components of the convective and the pressure gradient terms, respectively, of the momentum equation that cannot be captured by the finite element mesh. A key idea of the proposed method is that the convective subscale is close to a solenoidal field and the pressure gradient subscale is close to a potential field. The method ensures stability in anisotropic space–time discretizations, which is proved using numerical analysis for a linearized problem and demonstrated in classical numerical tests. The work includes a detailed description of the proposed formulation and several numerical examples that serve to justify our claims. •A new dynamic term-by-term stabilized finite element formulation is proposed.•The proposed method ensures stability in anisotropic space–time discretizations.•The dissipative structure of the method is analyzed and some stability results are presented.•Several numerical examples are presented to justify our claims.