Construction and application of algebraic dual polynomial representations for finite element methods on quadrilateral and hexahedral meshes

Estimation and numerical validation of inf-sup constant for bilinear form (p, div u)

“…In this paper, we derive a theoretical estimate for the discrete inf-sup formulation and validate the value of the constant using finite dimensional spaces defined in [1]. The theoretical proof of the inf-sup term becomes straight forward when we use the appropriate norm on K ⊥ space, see (5).…”

“…In this work we measure the norm of the velocity field in the space orthogonal to kernel of the divergence operator and prove that the value of the inf-sup constant β = 1.0. For numerical tests we use finite dimensional spaces defined in [1]. The validation study is performed on three different domains: i) unit square Ω = [0, 1] 2 ; ii) square Ω = [0, 2] 2 , for varying mesh sizes, h, and polynomial degrees N = 1, 2, 3.…”

“…We prove that the value of inf-sup constant is equal to 1.0 in all cases and is independent of the size and shape of the domain. Numerical tests for validation of inf-sup constants is performed using finite dimensional spaces defined in [1] on two test domains i) a square of size Ω = [0, 1] 2 , ii) a square of size Ω = [0, 2] 2 , for varying mesh sizes and polynomial degrees. The numeric values are in agreement with the theoretical value of inf-sup term.…”

A mass-, kinetic energy- and helicity-conserving mimetic dual-field discretization for three-dimensional incompressible Navier-Stokes equations, part I: Periodic domains

“…In this paper, u 1 and u 2 (ω 1 and ω 2 ) are approximated in different function spaces, and, therefore, equality between them will not hold unless the flow is fully resolved. Based on the promising results reported in this paper, we want to apply the algebraic dual polynomial spaces, [95], such that solutions u h 1 and u h 2 (ω h 1 and ω h 2 ) are two representations in a pair of algebraic dual polynomial spaces. As a result, we expect the difference between u h 1 and u h 2 (ω h 1 and ω h 2 ) to be smaller and using the vorticity from the other subset of equations, see (26) and (27), to be more consistent.…”

Use of algebraic dual representations in domain decomposition methods for Darcy flow in 3D domain

“…The primary objective of this paper is to address the first challenge, i.e. to define the framework and extend the use of algebraic dual representations introduced in [13] for DD formulation of Darcy flow. The DD formulation used in this work is based on the hybrid method which is a form of discontinuous Galerking formulation.…”