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Abstract: The aim of this paper is to study a finite difference method for quasilinear coupled problems of partial differential equations that presents numerically an unexpected second order convergence rate. The error analysis presented allows us to conclude that the finite difference method is supraconvergent. As the method studied in this paper can be seen as a fully discrete piecewise linear finite element method, we conclude the supercloseness of our approximations. [Copyright &y& Elsevier]

• Study of a system of quasilinear PDEs that can be used to model drug release from a polymeric structure enhanced by light. • A numerical scheme to compute the numerical solution of the IBVP based on the system of PDEs is proposed. • The theoretical support (stability and convergence) under weaker assumptions than those usually used is developed. • Numerical results illustrating the theoretical results established are provided. • A comparation between experimental data and simulation results is also presented. In this paper, we analyze a numerical scheme for a nonlinear coupled system of partial differential equations. Our study is motivated by the mathematical modeling of light-triggered drug delivery, a technique that can have a significant impact on cancer treatment. The numerical scheme combines a finite difference method (FDM) in space with an implicit-explicit (IMEX) method in time. For the main variable of interest - free drug concentration - we prove that the scheme is second-order supraconvergent in space in a discrete H 1 -norm and first-order convergent in time in a discrete L 2 -norm. We give numerical results illustrating the theoretical findings and computational simulations based on a laboratory experiment concerned with light-triggered drug delivery. [ABSTRACT FROM AUTHOR]

Convection enhanced drug delivery (CED) is a technique used to make therapeutic agents reach, through a catheter, sites of difficult access. The name of this technique comes from the convective flow originated by a pressure gradient induced at the tip of the catheter. This flow enhances passive diffusion and allows a more efficient spread of the agents by the target site. CED is particularly useful in the treatment of diseases that affect the central nervous system, where the blood–brain barrier prevents the diffusion of most therapeutic agents from the cerebral blood vessels to the brain interstitial space. In this work we deal with the numerical analysis of a coupled system of partial differential equations that can be used to simulate CED in an elastic medium like brain tissue. The model variables are the fluid velocity, the pressure, the tissue deformation, and the agents concentration. We prove the stability of the coupled problem and from the numerical point of view we propose a fully discrete piecewise linear finite element method (FEM). The convergence analysis shows that the method has second order convergence for the pressure, displacement, and concentration. Numerical experiments illustrating the theoretical convergence rates and the behavior of the system are also given. [ABSTRACT FROM AUTHOR]

In this paper, we solve a non-linear reaction–diffusion system with Dirichlet–Neumann mixed boundary conditions using a finite difference method (FDM) in space and the implicit midpoint method in time. This type of system appears, e.g., in the mathematical modeling of light-controlled drug delivery. One of the key results of this paper is the proof that the method has superconvergence second-order in space in a discrete H 1 -norm and optimal second-order convergence in time in a discrete L 2 -norm. Our result relies on the direct analysis of a suitable error equation, avoiding the classic construction of consistency plus stability implies convergence. One advantage of such an analysis technique is the establishment of the method's non-linear stability in an elegant way. Numerical examples support the theoretical convergence result. [ABSTRACT FROM AUTHOR]

In this paper we propose a numerical scheme for wave type equations with damping and space variable coefficients. Relevant equations of this kind arise for instance in the context of Maxwell's equations, namely, the electric potential equation and the electric field equation. The main motivation to study such class of equations is the crucial role played by the electric potential or the electric field in enhanced drug delivery applications. Our numerical method is based on piecewise linear finite element approximation and it can be regarded as a finite difference method based on non-uniform partitions of the spatial domain. We show that the proposed method leads to second order convergence, in time and space, for the kinetic and potential energies with respect to a discrete L 2 -norm. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we study numerical methods for integro-differential initial boundary value problems that arise, naturally, in many applications such as heat conduction in materials with memory, diffusion in polymers and diffusion in porous media. Here, we propose finite difference methods to compute approximations for the continuous solutions of such problems. We analyze stability and study convergence for those methods. Supraconvergent estimates are obtained. As such methods can be seen as lumped mass methods, our supraconvergent result corresponds to a superconvergent property in the context of finite element methods. Numerical results illustrating the theoretical results are included. [ABSTRACT FROM AUTHOR]