On a Framework for the Stability and Convergence Analysis of Discrete Schemes for Nonstationary Nonlocal Problems of Parabolic Type

Abstract

The main aim of this article is to propose a general framework for the theoretical analysis of discrete schemes used to solve multi-dimensional parabolic problems with fractional power elliptic operators. This analysis is split into three parts. The first part is based on techniques well developed for the solution of nonlocal elliptic problems. The obtained discrete elliptic operators are used to formulate semi-discrete approximations. Next, the fully discrete schemes are constructed by applying the classical and robust approximations of time derivatives. The existing stability and convergence results are directly included in the new framework. In the third part, approximations of transfer operators are constructed by using uniform and the best uniform rational approximations. The stability and accuracy of the obtained local discrete schemes are investigated. The results of computational experiments are presented and analyzed. A three-dimensional test problem is solved. The rational approximations are constructed by using the BRASIL algorithm.

Numerical Simulation of Fractional Power Diffusion Biosensors

“…Its implementation is efficient if all coefficients d j are non-positive. The results of stability and accuracy analysis obtained in [7] confirm that this requirement is satisfied for the given class of equations when the BRASIL algorithm is used to define rational approximations.…”

“…In order to solve it we apply the algorithm which gives an approximation of the solutions of the nonlocal diffusion equations (5.1)-(5.2). This algorithm uses the method which is proposed in [6,7]. It is based on rational approximations of nonlocal operators.…”

“…A very accurate approximation r M of the required rational function r M can be computed by applying the BRASIL algorithm, which is based on the barycentric rational formula [17]. In computational experiments done in [7] we used a free and open-source implementation of this algorithm in Python. We note that there are more implementations for the construction of BURA approximations, see e.g.…”

On a Stability of Non-Stationary Discrete Schemes with Respect to Interpolation Errors

“…As a basic technique for the stability analysis of discrete schemes proposed in this paper, we use the spectral method. It was efficiently used for non-stationary problems with nonlocal fractional-order elliptic operators [16,28,29].…”

“…The general convergence framework states that the stability and approximation properties guarantee the convergence of discrete solutions [1,2]. The deep, broad, and constructive theories of approximation and stability are developed, and they cover various important topics dealing with non-smooth data [3][4][5], weak solutions [6][7][8], energy and maximum principle stability estimates [1,2,9,10], ill-posed and inverse problems [11,12], and nonlocal mathematical models including fractional derivatives [13][14][15][16].…”