Well-posedness of an electric interface model and its finite element approximation

Abstract

This work aims at providing a mathematical and numerical framework for the analysis on the effects of pulsed electric fields on biological media. Biological tissues and cell suspensions are described as having a heterogeneous permittivity and a heterogeneous conductivity. Well-posedness of the model problem and the regularity of its solution are established. A fully discrete finite element scheme is proposed for the numerical approximation of the potential distribution as a function of time and space simultaneously for an arbitrary shaped pulse, and it is demonstrated to enjoy the optimal convergence order in both space and time. The presented results and numerical scheme have potential applications in the fields of medicine, food sciences, and biotechnology.Mathematics Subject Classification: 65M60, 78M30. The work of this author was substantially supported by Hong Kong RGC grants (projects 405513 and 404611). (zou@math.cuhk.edu.hk).Therefore, the function u := w + v is the desired weak solution.

On higher regularity for the Westervelt equation with strong nonlinear damping

“…assume that they are C 1,1 regular. The proof will expand on the approach taken in Lemma 3.6, [2]. Theorem 2.…”

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

“…Let Ω be a bounded domain in ℝ 2 with boundary ∂ Ω occupied by the concerned physical media having conductivity σ = σ ( x ) and permittivity ε = ε ( x ). Mathematically, electric voltage u is a solution to the time‐dependent equation . with initial and boundary conditions where f is the electric pulse of the model.…”

“…The performance of such kind of interface‐fitted FEMs depends not only on the quality of underlying finite element partition but also on the variational formulation of the problem. While the flux discontinuity of the solution can be captured in a variational formulation, the discontinuity of the solutions neither fit in the variational formulation nor satisfied in classical FEM solution spaces (see , and references therein). Many efforts have been made to develop alternative FEMs based on unfitted meshes for solving interface problems.…”

“…Equation (1.1) is numerically interesting as it does not belong to the well‐studied classes of time‐dependent equations. One of the first FEMs treating electric interface problem has been studied by Ammari et al . Well‐posedness of the model interface problem and the regularity of its solutions have been established.…”

“…Optimal convergence order in both L 2 ( H 1 ) and L 2 ( L 2 ) norms have been obtained with homogeneous jump conditions. In , authors have used fitted finite element discretization. More recently, L ∞ ( H 1 )‐norm and L ∞ ( L 2 )‐norm error estimates are derived in assuming solutions are continuous along interface.…”