Weak Galerkin finite element methods for electric interface model with nonhomogeneous jump conditions

Abstract

In this paper, the weak Galerkin finite element method (WG-FEM) is applied to a pulsed electric model arising in biological tissue when a biological cell is exposed to an electric field. A fitted WG-FEM is proposed to approximate the voltage of the pulsed electric model across the physical media involving an electric interface (surface membrane), and heterogeneous permittivity and a heterogeneous conductivity. This method uses totally discontinuous functions in approximation space and allows the usage of finite element partitions consisting of general polygonal meshes. Optimal pointwise-in-time error estimates in L 2 -norm and H 1 -norm are shown to hold for the semidiscrete scheme even if the regularity of the solution is low on the whole domain. Furthermore, a fully discrete approximation based on backward Euler scheme is analyzed and related optimal error estimates are derived. KEYWORDSinterface, optimal error estimates, pulsed electric field, weak Galerkin method 1

Finite volumes method for the cell electrical model

“…with the jump conditions (7). Also, ε ri = ε r + ∆tσ corresponds to the generalized permittivity for each media.…”

“…To solve our problem (5) under the boundary conditions (7), we modify the algorithm to take into account both the jump conditions on ε between the medium and on σ s at the surface. Only a discontinuity in the primal mesh is taken into account, since we adapt the mesh to the physic of the problem.…”

“…Figure 1. Cell model [7] The cytoplasm, media Ω c , is surrounded by the membrane, media Ω m . There is a jump condition between both mediums.…”

“…Nevertheless models are mostly based on numerical methods [5]. The majority of these methods involve a finite elements (FE) technique [5,7] and interpolation or refinement on the cell membrane [4].…”

A stabilizer free weak Galerkin finite element method for second‐order Sobolev equation

“…Previous work (cf. [34]) on WG-FEMs for pulsed electric field model problem is concerned only on first order time discretization. Present analysis provides a scope for the extension of these works to higher order time discretization method for electric interface model problem with polygonal meshes (cf.…”