Symbolic-Numerical Algorithms for Solving Multidimensional Boundary Value Problems by Finite Element Method on Hypercubes

Third- and fourth-order FEM schemes with multivariate Hermite interpolation polynomials of a d-dimensional hypercube for solving boundary value problems (BVPs) on hyperparallelepipedal meshes are elaborated. An exactly solvable model of a system of several identical particles with a pair oscillator interaction known as the Moshinsky atom is used as a test example. To describe the degenerate energy spectra of symmetric and antisymmetric bound states, the 2-, 3-, 4-, and 5- dimensional BVPs with Dirichlet and Neumann boundary conditions on a nonrectangular domain are formulated. To generate new FEM schemes with mixed partial derivatives, additional affine coordinate transformations are applied. Benchmark calculations of the BVPs confirm the order of declared FEM schemes. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2026.