A Maple Implementation of the Finite Element Method for Solving Boundary-Value Problems for Systems of Second-Order Ordinary Differential Equations

Abstract

We present a new algorithm of the finite element method (FEM) implemented as KANTBP 5M code in MAPLE for solving boundary-value problems (BVPs) for systems of second-order ordinary differential equations with continuous or piecewise continuous real or complex-valued coefficients. The desired solution in a finite interval of the real-valued independent variable is subject to mixed homogeneous boundary conditions (BCs). To reduce a BVP or a scattering problem with different numbers of asymptotically coupled or entangled open channels in the two asymptotic regions to a BVP on a finite interval, the asymptotic BCs for large absolute values of the independent variable are approximated by homogeneous Robin BCs. The BVP is discretized by means of the FEM using the Hermite interpolation polynomials with arbitrary multiplicity of the nodes, which preserves the continuity of derivatives of the desired solutions. The relevant algebraic problems are solved using the built-in linear algebra procedures. To calculate metastable states with complex eigenvalues of energy or to find bound states with the BCs depending on a spectral parameter, the Newton iteration scheme is implemented. Benchmark examples of the code application to BVPs and scattering problems of quantum mechanics are given.

V. Gerdt—It is painful to think Professor V. Gerdt is no longer among us, and this paper is his last contribution to development of solving BVPs on the base of the FEM which owes remarkable results to him. We are deeply grateful to him for his intuition, insight and support, which were invaluable during our long-standing collaboration.

Appendices

A Generation of IHPs on the Standard Interval

This appendix presents an algorithm for constructing IHPs according to their characteristics: p is the number of partitions of a finite element, $z_r$ are the IHP nodes with multiplicities $\kappa _r^{\max }$. They are applied to construct IHPs in the FEM scheme, then the conditions $z_r\in [0,1]$, $z_0=0$, $z_p=1$, $\kappa _0^{\max }=\kappa _p^{\max }$ are to be satisfied. For further implementation it is convenient to number IHPs with $n''$.

B Calculation of the FEM Scheme Characteristics

Note that when calculating the matrices (9) of the algebraic problem (8), we do it without explicitly calculating $N_s (z)$ from (7) by introducing global numbering $\varphi _r^\kappa$ on each of the finite elements $\varDelta _j$, i.e. $\varphi _S \equiv \varphi _ {n''} (z \in \varDelta _j) \equiv \varphi _r ^ \kappa (z \in \varDelta _j)$. In our implementation, the FEM IHP schemes are numbered so that S increases with an increase in j, or with a constant j and an increase in $n ''$, or with constant j and $n''$ and an increase in i. For convenience, arrays of length $n \times 3$ are introduced: E(j, 1) is the minimum S at which $\varphi _S$ is defined on $\varDelta _j$, E(j, 2) is the minimum S for which $r = p$ and $\varphi _S$ is defined on $\varDelta _j$, E(j, 3) is the maximum S at which $\varphi _S$ is defined on $\varDelta _j$ and a two-dimensional array C with dimension $S^{\max } \times 3$, where depending on S, C(S, 1) , C(S, 2) , C(S, 3) correspond to $\mu$ (the number of element of eigenvector $\boldsymbol{\varPhi }^ h$), $n''$ (the number of IHP) and i (the number of equation in the system of ODEs from Eq. (1)).

C FEM generation of Algebraic Eigenvalue Problem