Past Event: Oden Institute Seminar

A Semi-Hybrid approximation of the Stokes equations using H(div) spaces

Prof. Philippe Devloo, U Campinas, Brasil

Abstract

 One of the main challenges in proposing a numerical scheme for approximating the Stokes equations if the imposition of the incompressibility (i.e. divergence) constraint. The most popular approximation scheme for the Stokes equations, the Taylor Hood velocity/pressure combination is numerically stable, but fails to satisfy the divergence constraint in a pointwise or elementwise sense. 

H(div) approximation spaces have the advantage that, through the Piola transformation, they can be perfectly paired with a pressure lagrange multiplier space. In a first approach, the H(div)/pressure spaces were used in a discontinuous Galerkin context. This approach yielded the expected convergence rates for the velocity and pressure but resulted in a very wide connectivity stencil and therefore in an inneficient system inversion.

In this work, the standard H(div) space is used to approximate the velocity. The continuity of the tangent component of the velocity is imposed weakly using lagrange multipliers. The approximation scheme is denominated semi-hybrid because the approximation is continuous in the normal component of the velocity and the continuity of the tangential component of the velocity is imposed weakly. Stability is obtained by including higher order H(div) bubble functions. H(div) spaces of uniform polynomial order would lead to tangent traction lagrange multipliers that overconstrain the approximation.

The resulting numerical method has a favorable band structure and is optimally convergent. The internal velocity degrees of freedom can be condensed onto the boundary velocities and tangent traction.

An advantage of this choice of approximation space is that the coupling of the Stokes and Darcy equations can be easily represented. In the Darcy part of the domain, the weak continuity of the tangent velocity can be ommitted. The Brinkman equations can also be efficiently approximated by the proposed numerical scheme.

Biography

Prof. Devloo studied his undergraduate studies in Gent, Belgium in Electromechanical Engineering (1976 - 1981). After a one-year study of Computer Science in 1981, he enrolled at the University of Texas in Austin to pursue a PhD in computational mechanics under the supervision of J T Oden (1982 - 1987). From 1988-1922 worked at INPE (Space Research Institute in Brasil). From 1992-now is full professor at the State University of Campinas. 

Prof. Devloo conducts research in computational mechanics: translate engineering problems into a mathematical framework and approximate the system of conservation laws with the most appropriate numerical scheme.

Some of Prof. Devloo’s relevant contributions are

• hp-adaptive finite element approximations for H1, H(div) and H(curl) approximations

• object oriented finite element programming, being the author of a public domain framework NeoPZ https://github.com/labmec/neopz 

• Locally conservative approximations for flow in porous media

• multi-scale approximations applied to flow in porous media and elasticity (2D and 3D)

• hp-adaptive SBFem approximations for the Poisson problem and elasticity problem

Prof. Devloo has developed research projects with different companies such as Embraer, Petrobras, Equinor, TotalEnergies, applying advance finite element techniques to the solution of practical problems. Prof. Devloo has published more than 60 papers in refereed journals and oriented 17 PhD students.