A Laguerre-Legendre spectral element method for the solution of partial differential equations on infinite domains: application to the diffusion of tumour angiogenesis factors

In this paper, the spectral-element method formulation is extended to deal with semi-infinite and infinite domains without any prior knowledge of the asymptotic behaviour of the solution. A general spectral-element method which combines finite elements with basis functions as Lagrangian interpolants of Legendre polynomials and infinite elements with basis functions as Lagrangian interpolants of Laguerre functions, whilst preserving the properties of spectral-element discretizations: diagonality of the mass matrix, conformity, sparsity, exponential convergence, generality, and flexibility is presented. The Laguerre-Legendre spectral-element method of lines is applied to an evolutionary reaction-diffusion equation describing the early stages of the diffusion of tumour angiogenesis factors into the surrounding host tissue.