We analyze the spatially semidiscrete piecewise linear finite element method for a nonlocal parabolic equation resulting from thermistor problem. Our approach is based on the properties of the elliptic projection defined by the bilinear form associated with the variational formulation of the finite element method. We assume minimal regularity of the exact solution that yields optimal order erro...

Abstract. Dense, non-aqueous phase liquids (DNAPLs) are common organic contaminants in subsurface environment. Once spilled or leaked underground, they slowly dissolved into groundwater and generated a plume of contaminants. In order to manage the contaminated site and predict the behavior of dissolved DNAPL in heterogeneous subsurface requires a comprehensive numerical model. In this work, the...

We study a generalized Crank–Nicolson scheme for the time discretization of a fractional wave equation, in combination with a space discretization by linear finite elements. The schemeuses a non-uniformgrid in time to compensate for the singular behaviour of the exact solution at t = 0. With appropriate assumptions on the data and assuming that the spatial domain is convex or smooth, we show th...

For the Schrödinger-Poisson system, we propose an ALmost EXact(ALEX) boundary condition to treat accurately the numerical boundaries. Being local in both space and time, the ALEX boundary conditions are demonstrated to be effective in suppressing spurious numerical reflections. Together with the Crank-Nicolson scheme, we simulate a resonant tunneling diode. The algorithm produces numerical resu...

We show that the Strang splitting method applied to a diffusion-reaction equation with inhomogeneous general oblique boundary conditions is of order two when diffusion solved Crank-Nicolson method, while reduction occurs in if using other Runge-Kutta schemes or even exact flow itself for part. prove these results source term only depends on space variable, an assumption which makes scheme equiv...

In this paper, a Crank–Nicolson finite difference scheme based on cubic B-spline quasi-interpolation has been derived for the solution of coupled Burgers equations with Caputo–Fabrizio derivative. The first- and second-order spatial derivatives have approximated by first second quasi-interpolation. discrete obtained in way constitutes system algebraic associated bi-pentadiagonal matrix. We show...

Three new fully implicit methods which are based on the (5,5) Crank–Nicolson method, the (5,5) N-H (Noye–Hayman) implicit method and the (9,9) N-H implicit method are developed for solving the heat equation in two dimensional space with non-local boundary conditions. The latter is fourth-order while the others are second-order. While the implicit methods developed here, like the scheme based on...

The construction of a hierarchy of high-frequency microlocal artificial boundary conditions for the two-dimensional linear Schrödinger equation is proposed. These conditions are derived for a circular boundary and are next extended to a general arbitrarily-shaped boundary. They present the features of being differential in space and non-local in time since their definition involves some tempora...

Over the past years mathematical models, based on experimental data from MRI and CT scans, have been well developed to simulate the growth of aggressive forms of malignant brain tumours. The tumour growth model we are considering here, apart from proliferation and diffusion, is being characterized by a discontinuous diffusion coefficient to incorporate the heterogeneity of the brain tissue. For...

In this paper we present a posteriori error estimators for the approximate solutions of linear parabolic equations. We consider discretizations of the problem by discontinuous Galerkin method in time corresponding to variant Crank-Nicolson schemes and continuous Galerkin method in space. Especially, £nite element spaces are permitted to change at different time levels. Exploiting Crank-Nicolson...