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Sparse spectral methods on new geometries for differential equations

   Department of Computer Science

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  Dr Marco Fasondini, Prof R Davidchack  No more applications being accepted  Funded PhD Project (Students Worldwide)

About the Project

Spectral methods are used to compute solutions to differential and integral equations. Typically, this is done by expansions in orthogonal basis functions on the entire domain of the problem, or on multiple subdomains in a spectral element (or high-p-finite element) method.

Traditional spectral methods are well known to have excellent convergence properties for analytic solutions (exponentially fast convergence). However, this comes at the expense of solving dense and ill-conditioned linear systems. In recent years, spectral methods have been devised that result in sparse and well-conditioned linear systems that can be solved with fast, optimal complexity algorithms. Furthermore, these sparse spectral methods have been extended from intervals to certain regions in 2D and 3D, for example triangles, balls, cones, disks, disk slices, trapeziums and spherical caps (see, for example, [1]). In fact, sparse spectral methods can be designed for any geometry described by an algebraic curve or surface (zero sets of multivariate polynomials), however in practice one requires an explicit family of orthogonal basis functions or a stable and efficient computational method for constructing the basis functions.


The aim of this project is to design sparse spectral methods for a set of geometries described by algebraic curves and surfaces. The project supervisor and his collaborators have recently constructed new orthogonal polynomial basis functions (OPs) on a class of algebraic curves in 1D with a stable algorithm that has linear complexity [2,3]. Their methodology can be extended to construct OPs on regions in 2D bounded by the same class of algebraic curves and their associated surfaces of revolution in 3D. These OPs will be used in this project to construct sparse matrix representations of conversion (change-of-basis), multiplication and differentiation operators as well as transforms based on quadrature. These sparse matrices and transforms will be used to devise sparse spectral methods on the above-mentioned regions in 2D and 3D, which will be implemented in the open source Julia programming language. The resulting spectral methods will be tested on model problems that arise in acoustics and fluid mechanics (e.g., Laplace and Helmholtz problems). Ultimately, the goal is to extend these spectral methods to sparse spectral element methods for computationally challenging applications in numerical weather prediction, acoustic and elastic wave propagation and medical imaging.

Start date Sept 2023


UK and International* applicants are welcome to apply.

Entry requirements:

Applicants are required to hold/or expect to obtain a UK Bachelor Degree 2:1 or better in a relevant subject or overseas equivalent.  

The University of Leicester English language requirements may apply.

To apply

Please refer to the information and How to Apply section on our web site

Please ensure you include the project reference, supervisor and project title on your application.

Funding Notes

Future 50 Scholarship
Studentships provide funding for 3.5 years to include:
• Tuition fees at UK rates
• Stipend at UKRI rates for 2023 to be confirmed in early 2023 (currently £17,668 for 2022 entry)
• Access to a Research Training Support Grant of up to £1,500 pa for 3 years.
• Bench fees of £5,000 per annum for three years for laboratory-based studies
International applicants will need to be able to fund the difference between UK and International fees for the duration of study.


1] B. Snowball and S. Olver. Sparse spectral and p-finite element methods for partial differential equations on disk slices and trapeziums. Stud. Appl. Math., 145:3–35, 2020.
[2] M. Fasondini, S. Olver, and Y. Xu. Orthogonal polynomials on planar cubic curves. Found. Comput. Math., 1-31, 2021.
[3] M. Fasondini, S. Olver, and Y. Xu. Orthogonal polynomials on a class of planar algebraic curves. Submitted, see
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