Dual formulations of mixed finite element methods with applications

Abstract

Mixed finite element methods solve a PDE using two or more variables. The theory of Discrete Exterior Calculus explains why the degrees of freedom associated to the different variables should be stored on both primal and dual domain meshes with a discrete Hodge star used to transfer information between the meshes. We show through analysis and examples that the choice of discrete Hodge star is essential to the numerical stability of the method. Additionally, we define interpolation functions and discrete Hodge stars on dual meshes which can be used to create previously unconsidered mixed methods. Examples from magnetostatics and Darcy flow are examined in detail.

Low-order reconstruction operators on polyhedral meshes: application to compatible discrete operator schemes

“…These reconstruction functions are built using the Courant hat functions for potentials, the (lowest-order) Nédélec shape functions for circulations, and the (lowest-order) Raviart-Thomas-Nédélec shape functions for fluxes. A typical way to extend the reconstruction of potentials to polyhedral meshes is to use the concept of generalized barycentric coordinates; see Floater et al (2005), Gillette and Bajaj (2011), Gillette et al (2012), Hormann and Sukumar (2008), Wachspress (1975), Warren et al (2007) and references therein.…”

Numerical convergence of discrete exterior calculus on arbitrary surface meshes

“…The primal edge [5,1] is shared by the non-Delaunay pair [0, 5, 1] and [1,5,6]. According to edge [5,1] orientation with respect to both triangles, the dual edge [5,1] is defined to point from c 156 to c 051 , where c ijk = [i, j, k] is the node dual to the triangle [i, j, k]. Such an orientation would make [5,1] to be oriented 90 • counterclockwise with respect to [5,1] orientation in case the neighbor triangles were Delaunay.…”

“…According to edge [5,1] orientation with respect to both triangles, the dual edge [5,1] is defined to point from c 156 to c 051 , where c ijk = [i, j, k] is the node dual to the triangle [i, j, k]. Such an orientation would make [5,1] to be oriented 90 • counterclockwise with respect to [5,1] orientation in case the neighbor triangles were Delaunay. However, since triangles [0, 5,1] and [1,5,6] are non-Delaunay, the dual edge [5,1] has a flipped direction and is oriented 90 • clockwise with respect to edge [5,1] orientation.…”

“…Some properties, such as kinetic energy conservation for Euler equation, may require in addition the construction of a conservative time discretization as well. DEC was applied during the past decade to numerically solve various physical problems including Darcy [4,5] and incompressible Navier Stokes flows [6,7,3].…”

On the construction of general cubature formula by flat extensions

“…Applications of such algorithms determining cubature rules and cubature points over general domains occur in isogeometric modeling and finite element analysis using generalized Barycentric finite elements [17,1,35,36]. Additional applications abound in numerical integration for low dimensional (6–100 dimensions) convolution integrals that appear naturally in computational molecular biology [3,2], as well in truly high dimensional (tens of thousands of dimensions) integrals that occur in finance [32,8].…”