Abstract
SummaryIn this work we consider the numerical solution of incompressible flows on two‐dimensional manifolds. Whereas the compatibility demands of the velocity and the pressure spaces are known from the flat case one further has to deal with the approximation of a velocity field that lies only in the tangential space of the given geometry. Abandoning H 1‐conformity allows us to construct finite elements which are—due to an application of the Piola transformation—exactly tangential. To reintroduce continuity (in a weak sense) we make use of (hybrid) discontinuous Galerkin techniques. To further improve this approach, H(divΓ)‐conforming finite elements can be used to obtain exactly divergence‐free velocity solutions. We present several new finite element discretizations. On a number of numerical examples we examine and compare their qualitative properties and accuracy.
Highlights
Partial Differential Equations (PDEs) that are posed on curved surfaces play an important role in several applications in engineering, physics and mathematics
In this work we consider vector valued PDEs for viscous incompressible flows on surfaces that are immersed in the three dimensional space
In the context of discontinuous Galerkin (DG) discretizations Cockburn et al 41 were the first to realize that energy stability and local mass conservation for DG methods is only achieved for-conforming finite elements which result in pointwise divergence-free solutions. We extended this idea to hybrid DG (HDG) methods and considered several extensions and improvements and evaluated the computational efficiency of the resulting methods in Refs. 42,43,44,45,46,47,48, cf. the discussion in Section 3.5 below
Summary
Partial Differential Equations (PDEs) that are posed on curved surfaces play an important role in several applications in engineering, physics and mathematics. In this work we follow a different approach: we abandon continuity of the finite elements. This loss of conformity allows us to construct exactly tangential vector fields. This is achieved by mapping finite element functions from the two-dimensional reference element by a straight-forward generalization of the well-known Piola transformation. This guarantees that the resulting (possibly higher order) vectorial basis functions are exactly tangential to the surface. To deal with the missing continuity we apply well established techniques from the flat case: discontinuous Galerkin (DG) methods and variants such as the hybrid DG (HDG) methods
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