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This paper is concerned with the numerical solution of high-dimensional Fokker-\ud Planck equations related to multi-dimensional diffusion with polynomial coefficients\ud or Pearson diffusions. Classification of multi-dimensional Pearson diffusion follows\ud from the classification of one-dimensional Pearson diffusion. There are six important\ud classes of Pearson diffusion - three of them possess an infinite system of moments\ud (Gaussian, Gamma, Beta) while the other three possess a finite number of moments\ud (inverted Gamma, Student and Fisher-Snedecor). Numerical approximations to the\ud solution of the Fokker-Planck equation are generated using the spectral method.\ud The use of an adaptive reduced basis technique facilitates a significant reduction in\ud the number of degrees of freedom required in the approximation through the determination\ud of an optimal basis using the singular value decomposition (SVD). The\ud basis functions are constructed dynamically so that the numerical approximation\ud is optimal in the current finite-dimensional subspace of the solution space. This is\ud achieved through basis enrichment and projection stages. Numerical results with different\ud boundary conditions are presented to demonstrate the accuracy and efficiency\ud of the numerical scheme.

New spectral element basis functions are constructed for problems possessing an axis of symmetry. In problems defined in domains with an axis of symmetry there is a potential problem of degeneracy of the system of discrete equations corresponding to nodes located on the axis of symmetry. The standard spectral element basis functions are modified so that the axial conditions are satisfied identically. The modified basis is employed only in spectral elements that are adjacent to the axis of symmetry. This modification of the spectral element method ensures that the nodes are the same in each element, which is not the case in other methods that have been proposed to tackle the problem along the axis of symmetry, and that there are no nodes along the axis of symmetry. The problems of Stokes flow past a confined cylinder and sphere are considered and the performance of the original and modified basis functions are compared.

The governing equations for Stokes flow are formulated in terms of a stream function and Airy stress function. This formulation ensures that mass and momentum are conserved identically. In terms of these new variables, the equations of motion are written as a second-order elliptic system. These equations are embedded in biharmonic equations and the boundary conditions appropriate for this higher-order system are determined using a least-squares process. This technique is applied to the planar stick-slip problem. A numerical solution to the problem is obtained using a spectral domain decomposition method. An algebraic mapping is used to treat the flow domain without truncation. The coefficients in a singular expansion of the stream function about the stick-slip singularity are computed using a post-processing technique.

A spectral element method is described which enables Poisson problems defined in irregular infinite domains to be solved as a set of coupled problems over semi-infinite rectangular regions. Two choices of trial functions are considered, namely the eigenfunctions of the differential operator and Chebyshev polynomials. The coefficients in the series expansions are obtained by imposing weak C1 matching conditions across element interfaces. Singularities at re-entrant corners are treated by a post-processing technique which makes use of the known asymptotic behaviour of the solution at the singular point. Accurate approximations are obtained with few degrees of freedom.

This paper focuses on Pearson diffusions and the spectral high-order approximation of their related Fokker–Planck equations. The Pearson diffusions is a class of diffusions defined by linear drift and quadratic squared diffusion coefficient. They are widely used in the physical and chemical sciences, engineering, rheology, environmental sciences and financial mathematics. In recent years diffusion models have been studied analytically and numerically primarily through the solution of stochastic differential equations. Analytical solutions have been derived for some of the Pearson diffusions, including the Ornstein–Uhlenbeck, Cox–Ingersoll–Ross and Jacobi processes. However, analytical investigations and computations for diffusions with so-called heavy-tailed ergodic distributions are more difficult to perform. The novelty of this research is the development of an accurate and efficient numerical method to solve the Fokker–Planck equations associated with Pearson diffusions with different boundary conditions. Comparisons between the numerical predictions and available time-dependent and equilibrium analytical solutions are made. The solution of the Fokker–Planck equation is approximated using a reduced basis spectral method. The advantage of this approach is that many models for pricing options in financial mathematics cannot be expressed in terms of a stochastic partial differential equation and therefore one has to resort to solving Fokker–Planck type equations.

The effectiveness of relaxation schemes for solving the systems of algebraic equations which arise from spectral discretizations of elliptic equations is examined. Iterative methods are an attractive alternative to direct methods because Fourier transform techniques enable the discrete matrix-vector products to be computed almost as efficiently as for corresponding but sparse finite difference discretizations. Preconditioning is found to be essential for acceptable rates of convergence. Preconditioners based on second-order finite difference methods are used. A comparison is made of the performance of different relaxation methods on model problems with a variety of conditions specified around the boundary. The investigations show that iterations based on incomplete LU decompositions provide the most efficient methods for solving these algebraic systems.