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Funding and scholarships for students

Award details

Numerical analysis of innovative algorithms for solving partial differential equations applied to complex, multiscale problems. EMPS College Home fees Studentship, PhD in Mathematics. Ref: 4310

About the award


Lead Supervisor:

Dr Jemma ShiptonCollege of Engineering, Mathematics and Physical Sciences, Streatham Campus, University of Exeter




Department of Mathematics, Streatham Campus, Devon, University of Exeter

The University of Exeter’s College of Engineering, Mathematics and Physical Sciences  is inviting applications for a fully-funded PhD studentship to commence in January 2022 or as soon as possible thereafter. The studentship will cover Home tuition fees plus an annual tax-free stipend of at least £15,609 for 3.5 years full-time, or pro rata for part-time study. 

This College studentship is open to UK and Irish nationals, who if successful in their application will receive a full studentship including payment of university tuition fees at the home fees rate.

Project Description:

The partial differential equations (PDEs) that govern fluid motion present many difficulties to numerical algorithms. In geophysical applications, such as modelling the Earth's atmosphere or oceans, rotation and stratification lead to large scale balances that must be maintained by the numerical algorithm. In astrophysical applications the presence of a magnetic field leads to additional waves. In biological applications the flow is affected by moving, elastic boundaries. In all of these applications, small scale features develop that have impacts of the larger scale flow, leading to complex multiscale problems that require efficient use of high performance supercomputers. The design and analysis of algorithms that can exploit the full power of supercomputers is an important and timely topic, due to the fact that the next generation of supercomputer architectures will be radically different, requiring orders of magnitude more parallelism (the ability to perform calculations simultaneously).

This project will focus on the use of finite element methods for fluid flows. Finite element methods represent fields on a discrete mesh made of individual cells, or elements. Within each element the field is approximated by a polynomial of a chosen degree, and fields can have different levels of continuity between the elements. These choices impact on the ability of the algorithm to maintain the properties of the flow and lead to many interesting research questions relating to the design of stable, accurate, robust, efficient and scalable solvers. In return, finite element methods provide the flexibility to represent complex domains using non-orthogonal grid with varying resolution that can even change as the flow evolves. This project will use the Firedrake finite element library, which greatly simplifies implementation of these schemes, enabling flexible and rapid investigation of a range of ideas.

Depending on the skills and interests of the successful applicant, potential projects include:

  • design, analysis and implementation of innovative algorithms for timestepping PDEs, including algorithms that introduce parallelism in the time domain
  • design, analysis and implementation of subgridscale parameterizations to better represent the impacts of small scale dynamics on large scale flow, including the use of parcel-in-cell methods and superparameterizations which require embedding ensembles of subgrid models within the larger simulation domain
  • application of finite element methods to complex flows to investigate topics such as:
    • the formation and modelling of fronts in the atmosphere
    • the stability of jets and vortices in the presence of forcing from physical processes, such as phase changes from moisture or the presence of magnetic fields
    • the impact of non-isotropic bladder elasticity on urinary flows

Entry requirements

This studentship is open to UK and Irish nationals, who if successful in their application will receive a full studentship including payment of university tuition fees at the home fees rate.

Applicants for this studentship must have obtained, or be about to obtain, a First or Upper Second Class UK Honours degree, or the equivalent qualifications gained outside the UK, in an appropriate area of science or technology.

Experience in computer programming, particularly using Python, would be advantageous but is not essential.

If English is not your first language you will need to have achieved at least 6.0 in IELTS and no less than 6.0 in any section by the start of the project. 

Alternative tests may be acceptable (see

How to apply

In the application process you will be asked to upload several documents.  Please note our preferred format is PDF, each file named with your surname and the name of the document, eg. “Smith – CV.pdf”, “Smith – Cover Letter.pdf”, “Smith – Transcript.pdf”.

• CV
• Letter of application (outlining your academic interests, prior project work experience and reasons for
  wishing to undertake the project).
• Transcript(s) giving full details of subjects studied and grades/marks obtained (this should be an
   interim transcript if you are still studying)
• Two references from referees familiar with your academic work. If your referees prefer, they can email
   the reference direct to quoting the studentship reference number.
• If you are not a national of a majority English-speaking country you will need to submit evidence of your proficiency in English.  Please see the entry requirements information above.

The closing date for applications is midnight on 24th January 2022.  Interviews will be held online on the week commencing 7th February 2022.

If you have any general enquiries about the application process please email

Project-specific queries should be directed to the main supervisor at


Application deadline:24th January 2022
Value:Home tuition fees plus an annual tax-free stipend of at least £15,609 for 3.5 years full-time, or pro rata for part-time study.
Duration of award:per year
Contact: PGR Admissions Office