Solution of Boundary-Value Problems using Kantorovich Method

A.A Gusev,L.L Hai,O Chuluunbaatar,S.I Vinitsky,V.L Derbov,Gh Adam,M Hnatič,J Buša

doi:10.1051/epjconf/201610802026

Abstract

We propose a computational scheme for solving the eigenvalue problem for an elliptic differential equation in a two-dimensional domain with Dirichlet boundary conditions. The solution is sought in the form of Kantorovich expansion over the basis functions of one of the independent variables with the second variable treated as a parameter. The basis functions are calculated as solutions of the parametric eigenvalue problem for an ordinary second-order differential equation. As a result, the initial problem is reduced to a boundary-value problem for a set of self-adjoint second-order differential equations for functions of the second independent variable. The discrete formulation of the problem is implemented using the finite element method with Hermite interpolation polynomials. The effciency of the calculation scheme is shown by benchmark calculations for a square membrane with a degenerate spectrum.

Highlights

We propose a computational scheme for solving the eigenvalue problem for an elliptic differential equation in a two-dimensional domain with Dirichlet boundary conditions
The solution is sought in the form of Kantorovich expansion over the basis functions of one of the independent variables with the second variable treated as a parameter
The initial problem is reduced to a boundary-value problem for a set of self-adjoint second-order differential equations for functions of the second independent variable

Summary

Introduction

The calculation of spectral and optical properties of electronic states in axially symmetric quantum dots is reduced to the solution of two-dimensional boundary-value problems (BVP) for elliptic differential equations with nonseparable variables in a finite domain [1]. For the impurity states of quantum dots such BVPs are defined in domains of complicated geometry and involve piecewise-continuous potential functions In this case it is necessary to preserve the continuity of the approximate solution, and the continuity of its first derivative, which is most naturally achieved using the finite element method with Hermite interpolating polynomials [6, 7]. We demonstrate the efficiency of the programs generated in Maple and Fortran for 100×100 and higher-order matrices, respectively, in benchmark calculations for the exactly solvable eigenvalue problem of a square membrane with degenerate spectrum. This example is not trivial from the computational view point. The use of new coordinates that can be separated within the domain but not at the boundary allows the simulation of a potential function, depending upon two variables, and justifies the application of the Kantorovich method

Kantorovich Method

Benchmark calculation: rectangular membrane