One Stroke Complementarity for Poisson-Like Problems

Abstract

Taking electrokinetics as a paradigm problem for the sake of simplicity, complementarity originates when an irrotational electric field and a solenoidal current density satisfying boundary conditions are in hand. We first compare three formulations to obtain a solenoidal current density, both in terms of pure computational advantage and in the ability to pursue symmetric energy bounds with respect to the standard electric scalar potential formulation. For these formulations, we devise post-processing techniques that promise to provide bilateral bounds in one stroke, hence requiring the solution of just one linear system.

Diagonal Discrete Hodge Operators for Simplicial Meshes Using the Signed Dual Complex

“…On the contrary, in this paper, we explore advantages in solving Poisson-like BVPs formulated as [26] where D contains the incidences between elements-faces pairs, V contains the unknown scalar potential on dual nodes, and U s is a known electromotive force used to impose Dirichlet boundary conditions. We compare this formulation D in terms of accuracy and simulation time with other four formulations surveyed in [26]: 1) the finite elements formulation V based on the scalar potential; 2) the formulation T based on the vector potential; 3) the geometric mixed-hybrid formulation H; and 4) theṼ based on a sparse construction of dual Hodge operators.…”

“…A further code validation is performed by verifying the convergence of the resistance of a square resistor [26] with an increasing number of elements (there is no room to show the results here). We present the results on a more complicated benchmark proposed in [5], which comprises a micro electro-mechanical system (MEMS) comb drive placed on a ground plane.…”

Explicit geometric construction of sparse inverse mass matrices for arbitrary tetrahedral grids

“…There is a second method to produce complementary formulations that we call complementary-dual : they are complementary formulation but still use the scalar potential U Ñ , which is sampled on grid dual nodes, one-to-one with grid cells. An effective complementary-dual formulation, which is algebraically equivalent to the V P formulation, is the mixed-hybrid M H formulation [30], [31].…”

An Arbitrary-Order Discontinuous Skeletal Method for Solving Electrostatics on General Polyhedral Meshes

“…where U contains square-integrable, finite energy functions which comply with the boundary condition (1c). At the discrete level, the local elimination of the displacement is performed by enforcing the continuity of interface DOFs in D k h by Lagrange multipliers (which can be interpreted as in [5] as traces of the potential) and inverting inside each element T P T h the local constitutive laws expressed by (6a); cf. [7] for the details.…”

“…The paper also emphasizes the analogies, in the lowest order case corresponding to the polynomial degree k " 0, with the Discrete Geometric Approach (DGA) [2]. In particular, this novel method is algebraically equivalent (up to the choice of a scalar parameter in the stabilisation term) to the mixed-hybrid geometric formulation described in [5]. The MHO method bears also some similarities with the Discontinuous Galerkin (DG) [6], but it presents a higher convergence rate for a given polynomial degree.…”