Virtual Element Method (VEM)

We propose to develop the Virtual Element Method (VEM), a novel paradigm for the numerical solution of Partial Differential Equations (PDEs).

The VEM is an extremely flexible mesh based method allowing for the use of general polygonal/polyhedral meshes including non-convex, degenerate, and non-matching elements. It is a unifying framework from which L2, H1, H2, H(div), and H(curl) elements can be naturally constructed. In particular, the VEM can deliver highly regular solutions, hence proving better suited than standard finite element techniques for the solution of higher-order problems and eigenvalues approximation.

The VEM trial functions are solutions of suitable partial differential equation problems inside each element, as within generalised Finite Elements [3]. The novelty of the VEM approach is that it avoids computing with the trial virtual functions by basing all computations solely on a set of carefully chosen degrees of freedom. In this way, the VEM achieves flexibility while maintaining the ease of implementation of classical FEMs.

Within this project:
- the VEM framework will be further and developed for the solution of problems driven by application in fluid dynamics and biomedicine,
- the flexibility of the VEM will be exploited for the first time for the design of multiscale approaches and mesh and order adaptive algorithms,
- new public domain software implementing the VEM on general meshes will be published and integrated into the Distributed and Unified Numerics Environment (DUNE).

The VEM is a new framework for the solution of partial differential equation (PDE) problem. A full range of L2, H1, H2, H(div), and H(curl) elements can be constructed, and easily computed with. It provides greater flexibility than standard Finite Element approaches in terms of mesh partitions and discretisation spaces.

The context is that of the ever growing number of generalised FEM and non-standard basis functions approaches such as GFEM, PoU, XFEM, MFEM, HMM, HMH, BEM, Trefftz, meshfree, PFEM, DPG, IGA, MFD, VEM, . . . ). The state-of-the-art of such activity will be assessed at the EPSRC funded 2014 LMS Durham Symposium "Building bridges: connections and challenges in modern approaches to numerical partial differential equations", co-organised by the PI, and to be held only few months after the start of this project.

Over the above mentioned techniques, the VEM has a clear advantage: it has a computational cost comparable to that of standard FEM. Nevertheless, as only linear diffusion and elasticity problems have been tackled so far, full capabilities of the VEM framework are still to be explored/exploited. Fundamental questions such as convergence and stability under collapsing mesh edges/faces have to be answered, that would give a definitive practical advantage of the VEM approach. Finally, more advance techniques such as the use of meshes with curved edges/faces, automatic mesh-adaptation, order-adaptation algorithm, and computing with moving meshes, which are all natural developments given the method's flexibility, have yet to be considered.

This project aims to give a fundamental contribution to the development of these techniques. This development will be directed and facilitated by a number of collaborations in the UK and abroad which are at the forefront of modern numerical analysis and PDEs solvers software development.

The proposed work will be disseminated to the wider audience through a dedicated project website, managed by the PI. This will hold the software produced within the project. In particular, codes for simulating molecular trafficking inside biological cells and coupled Darcy and Stokes flows in ground/surface flows and engineering filtration will be published. In order to ensure sustainability and availability to a larger audience, the software will be integrated into the Distributed and Unified Numerics Environment (DUNE), thus making available cutting edge scientific computing techniques to the big user community within the geosciences.

Collaboration with the British Geological Survey is ongoing, and contacts with the industry, namely Weatherford International and Schulumberger, are already being pursued in order to facilitate technology transfer. On the biological application side of the project, collaboration is ongoing with cell biologists (the group of A. Fry) within the University of Leicester, aimed at the discovery of rate-limiting steps of the signalling cascades as potential targets of pharmacological therapy.

This project will permit the PI to establish in the UK a research group leading the development of novel numerical techniques and their applications. The PI will acquire the skills to manage a research project, thus growing both as a scientist and manager of scientific advancement.

Within this project, a Research Assistant and a PhD student will be trained. They will be exposed to state-of-the-art numerical techniques and invited to collaborate with top numerical analysts and scientific computing software developers, both in the UK and internationally. Further, they will gain valuable programming and scientific computing skills in object programming, adaptivity, and the use of large parallel computing software.