An arbitrary-order method for magnetostatics on polyhedral meshes based on a discrete de Rham sequence

Abstract

In this work, we develop a discretisation method for the mixed formulation of the magnetostatic problem supporting arbitrary orders and polyhedral meshes. The method is based on a global discrete de Rham (DDR) sequence, obtained by patching the local spaces constructed in [Di Pietro, Droniou, Rapetti, Fully discrete polynomial de Rham sequences of arbitrary degree on polygons and polyhedra, arXiv:1911.03616] by enforcing the single-valuedness of the components attached to the boundary of each element. The first main contribution of this paper is a proof of exactness relations for this global DDR sequence, obtained leveraging the exactness of the corresponding local sequence and a topological assembly of the mesh valid for domains that do not enclose any void. The second main contribution is the formulation and well-posedness analysis of the method, which includes the proof of uniform Poincaré inequalities for the discrete divergence and curl operators. The convergence rate in the natural energy norm is numerically evaluated on standard and polyhedral meshes. When the DDR sequence of degree $k\ge 0$ is used, the error converges as $h^{k+1}$, with $h$ denoting the meshsize.

References in corpus (4)

• Lowest order Virtual Element approximation of magnetostatic problems
• H(div) and H(curl)-conforming VEM
• Fully discrete polynomial de Rham sequences of arbitrary degree on polygons and polyhedra
• A discrete Weber inequality on three-dimensional hybrid spaces with application to the HHO approximation of magnetostatics

Cited by in corpus (10)

• Arbitrary-order pressure-robust DDR and VEM methods for the Stokes problem on polyhedral meshes
• Fully discrete polynomial de Rham sequences of arbitrary degree on polygons and polyhedra
• A discrete Weber inequality on three-dimensional hybrid spaces with application to the HHO approximation of magnetostatics
• An arbitrary-order discrete de Rham complex on polyhedral meshes: Exactness, Poincaré inequalities, and consistency
• A polyhedral discrete de Rham numerical scheme for the Yang-Mills equations
• An exterior calculus framework for polytopal methods
• Cohomology of the discrete de Rham complex on domains of general topology
• Inverting the discrete curl operator: a novel graph algorithm to find a vector potential of a given vector field
• Uniform Poincaré inequalities for the Discrete de Rham complex on general domains
• An arbitrary-order discrete de Rham complex on polyhedral meshes. Part II: Consistency