Numerical Analysis

Numerical analysis refers to the development and implementation of algorithms for the efficient determination of computational solutions to mathematical problems. Efficiency can, depending on the context, mean algorithmic accuracy, stability, or speed; usually all three of these are important.

Research in numerical analysis at Bath encompasses a wide spectrum of modern topics. A connecting thread which runs through much of our research is the derivation, analysis and implementation of computational techniques for Partial Differential Equations (PDEs) of various types, or their reduction to a system of Differential Algebraic Equations (DAEs).

There is a weekly informal seminar series, followed by a research group lunch, along with regular research workshops organised with industry (for example the Rutherford Appleton Laboratory) and with other academic institutions. There is a weekly informal seminar series, followed by a research group lunch, along with regular research workshops organised with industry (for example the Rutherford Appleton Laboratory) and with other academic institutions. The Numerical Analysis group webpages, contain further details of current events.

Our research in numerical analysis can be arranged under the following broad headings:

• Nonlinear elliptic and parabolic PDEs
• Dynamical systems and DAEs
• Boundary element methods
• Numerical linear algebra and parallel algorithms
• Adaptive methods
• Numerical bifurcation theory

Related areas of our research include:

• Analysis and Differential Equations
• Industrial Applied Mathematics
• Continuum Mechanics of Solids and Fluids

Further details of specific research projects are given below.

Nonlinear elliptic and parabolic PDEs

Research staff: Chris Budd, Melina Freitag, Ivan Graham, Rob Scheichl

Problems of this nature arise frequently in mathematical models of physical processes such as combustion, pattern formation and buckling. To find numerical solutions is a challenging task both because of the inherent complexity of the problems and also because the interesting physical solutions often have singularities and/or interfaces.

One set of current investigations centres on finding efficient numerical methods which will accurately compute solutions with singularities in a variety of (three dimensional) geometries. This involves a mixture of analysis and algorithm design, often referring back to the underlying system we are modelling.

Dynamical systems and DAEs

Research staff: Chris Budd

Classical estimates of the error made by numerical solutions of initial value problems grow unboundedly with the time-interval under consideration. New techniques are therefore required to interpret the meaning of computations carried out over long time- intervals. Recent advances have been made in this area by using methods from abstract dynamical systems theory to consider the behaviour of the numerical approximation. Related work is also under way into the study of how to best approximate dynamical systems that preserve some special property; eg a system of differential equations where the solution is constrained to remain on the surface of a sphere. Work is in progress to compare the behaviour of classical high-order numerical solvers with methods specially designed to preserve the required property.

Other research involves the derivation and analysis of algorithms to capture long-time dynamical phenomena, for example periodic and homoclinic orbits, in large scale dynamical systems. Such systems arise in discretised time dependent partial differential equations and in such cases there is a subtle interplay between the numerical linear algebra and dynamical system requirements.

Boundary element methods

Research staff: Ivan Graham

These are popular techniques in Numerical Engineering for the computation of solutions of classical PDEs on complicated domains, especially in three dimensions. When they can be applied they offer a number of advantages over domain methods, since they reduce the problem to the lower dimensional boundary manifold and they can handle far field boundary conditions exactly. Recent work on three dimensional applications includes fast methods of assembling the corresponding stiffness matrices and solving the linear equations by iterative methods. Here focus has been on performance of various algorithms in the presence of degenerate (anisotropic) mesh refinement. Another project involves the computation of high frequency asymptotics in acoustic and electromagnetic scattering.

Numerical linear algebra and parallel algorithms

Research staff: Ivan Graham, Rob Scheichl, Melina Freitag

A recently very active area is in the parallelisation of iterative methods for solving algebraic equations arising from finite element discretisations of PDEs governing various applications such as Navier Stokes flow and groundwater flow. Research is concerned both with the rigorous analysis of the performance of algorithms (this involves finite element theory) and the implementation of algorithms on multiprocessor machines (this requires techniques from computer science and an interest in large scale programming). At present the work is concentrating on problems arising from unstructured meshes in three dimensions and on the solution of coupled systems with industrial applications.

Iterative methods are also employed to find a small number of important eigenvalues in large, sparse, generalised eigenvalue problems, often arising from finite element discretisations of partial differential equations. Current research centres on the analysis of iterative methods, like Arnoldi's method, applied to shifted systems with the linear solves implemented iteratively, say using GMRES.

Adaptive methods

Research staff: Chris Budd, Ivan Graham, Rob Scheichl

An adaptive numerical method is any method which adapts its solution procedure to take account of developing structure in the computed results. Such a process is frequently required when computing problems which have solutions that form singularities, sharp interfaces, have greatly varying time-scales or where spatial and temporal structures are closely coupled. Without some form of adaptivity it is possible for the numerical method to give completely wrong (spurious) solutions.

One body of work centres on studying finite difference and finite element methods for parabolic PDEs where the computational mesh is adopted. Exciting new results are obtained by using an adaptivity procedure based upon moving mesh partial differential equations with scaling invariance properties. Such methods afford the promise of reliably reproducing the dynamics of the underlying PDE, of preserving invariant manifolds and of having wide applicability. Much needs to be done, especially in extending these methods to systems and problems in higher dimensions.

Another area of current research is an investigation of how sensitive an elliptic problem can be to changes in the right-hand side and its boundary conditions. This work uses methods of classical analysis to estimate the norm of the solution operator of the problem, which gives a quantitative measure of the sensitivity. Such estimates can then be used to bound the numerical error made in approximating the elliptic problem.

Numerical bifurcation theory

Research staff: Alastair Spence

In many physical applications, stability is determined by evaluation of certain critical eigenvalues of a linearised discretised system as a parameter varies. Often these discretised systems can be very large (with over ten thousand degrees of freedom) and explicit calculation of all eigenvalues is impractical. Current research involves the development of efficient techniques for the detection of Hopf bifurcation points and bifurcation points in the presence of symmetries in discretised Navier-Stokes problems.