Numerical analysis of locally conservative weak Galerkin dual-mixed finite element method for the time-dependent Poisson–Nernst–Planck system

Convergence analysis of weak Galerkin flux-based mixed finite element method for solving singularly perturbed convection-diffusion-reaction problem

“…Proof. See [21,Lemma 4]. Now we are in position of establishing the main result of this section, namely, the suboptimal rate of convergence for the weak Galerkin scheme provided by Problem 3.…”

“…(cf. [28,36,36,29,31,32,11,25,27,12,21,13])), demonstrating their substantial potential as a formidable numerical method in scientific computing. The key distinction between WG methods and other existing finite element techniques lies in their utilization of weak derivatives and weak continuities in formulating numerical schemes based on conventional weak forms for the underlying PDE problems.…”

“…Despite the extensive research on the weak Galerkin FEM, there are few publications on the weak Galerkin mixed-FEM in the literature. Specifically, and up to our knowledge, [32] and [21] are the only two works in which the weak Galerkin mixed-FEM for linear and nonlinear PDEs based on the dual formulation are proposed and analyzed, including corresponding optimal errors estimates. Although many research works have been studied and analyzed on the weak Galerkin method for the Navier-Stokes problem (cf.…”

“…In fact, instead of using the primal or dual-mixed method (see e.g. [32,21]), we now employ a modified mixed formulation and adapt the approach from [22] to propose, up to our knowledge by the first time, a weak Galerkin mixed-FEM for Navier-Stokes, which consists of introducing the gradient of velocity and a vector version of the Bernoulli tensor as further unknowns. In this way, and besides eliminating the pressure, which can be approximated later on via postprocessing, the resulting mixed variational formulation does not need to incorporate any augmented term, and it yields basically the same Banach saddle-point structure.…”

“…3, following Refs. [32] and [21]. This section comprises four main parts: first, we state basic assumptions on the mesh; second, we define the local WG space, projections, and weak differential operators; third, we discuss their approximation properties; and fourth, we derive the global WG subspace and the disceret scheme.…”