A new approach to convergence analysis of linearized finite element method for nonlinear hyperbolic equation

Abstract

We study a new way to convergence results for a nonlinear hyperbolic equation with bilinear element. Such equation is transformed into a parabolic system by setting the original solution u as u t = q. A linearized backward Euler finite element method (FEM) is introduced, and the splitting skill is exploited to get rid of the restriction on the ratio between h and τ. The boundedness of the solutions about the time-discrete system in H 2-norm is proved skillfully through temporal error. The spatial error is derived without the mesh-ratio, where some new techniques are utilized to deal with the problems caused by the new parabolic system. The final unconditional optimal error results of u and q are deduced at the same time. Finally, a numerical example is provided to support the theoretical analysis. Here h is the subdivision parameter, and τ is the time step.

Simulation of the influence of voltage level and pulse spacing on the efficiency, aggressiveness and uniformity of the electroporation process in tissues using meshless techniques

“…To the best of our knowledge, meshless techniques have not been used before to simulate drug uptake in electroporated tissues. These techniques are advantageous compared to other meshed techniques commonly used for this kind of problem, such as FEM and FVM, in the sense that they do not require the use of a domain mesh but instead the use of collocation points in the boundary and the domain. Additionally, the high‐order approximations of the field variables and their derivatives, as well as the versatility of the application of boundary conditions in comparison with meshed methods, make these techniques very attractive.…”