L^∞(L²) and L^∞(H¹) norms error estimates in finite element methods for electric interface model

ABSTRACT

In this paper, we analyze finite element methods applied to pulsed electric model arising in biological tissue when a biological cell is exposed to an electric field. Considering the cell to be a conductive body, embedded in a more or less conductive medium, the governing system involves an electric interface (surface membrane), and heterogeneous permittivity and a heterogeneous conductivity. A fitted finite element method with straight interface triangles is proposed to approximate the voltage of the pulsed electric model across the physical media. Optimal pointwise-in-time error estimates in $L^2$-norm and $H^1$-norm are shown to hold for semidiscrete scheme even if the regularity of the solution is low on the whole domain. Further, a fully discrete approximation based on Crank–Nicolson scheme is analyzed and related optimal error estimates are derived.

KEYWORDS:

• Electric interface
• finite element method
• discontinuous coefficients
• semidiscrete scheme
• fully discrete scheme
• optimal error estimates

AMS SUBJECT CLASSIFICATIONS(2000):

• 65N15
• 65N30

Acknowledgments

The authors are grateful to the anonymous referees for their valuable comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the authors.