Noel J. Walkington
Numerical Approximation of Partial Differential Equations
- Mixtures of Fluids: The flow of incompressible, imiscible, fluids
(e.g. oil droplets in water) is modeled by the Navier Stokes (momentum) equation,
with discontinuous density and viscosity, coupled to a first order convection
equation representing the balance of mass. The key ingredient required to
establish existence of solutions to this system is a compactness result
of DiPerna and Lions for solutions of the convection equation with non-smooth
coefficients. Recently I developed a parallel theory for numerical approximations
of the convection equation. This result enabled C. Liu and myself to prove
convergence of numerical approximations of solutions to the coupled system
governing the flow of mixtures of imiscible fluids.
- Parabolic Problems and the Wasserstein Metric: Recently F. Otto
has shown that certain parabolic problems (e.g. the porous medium and Fokker
Planck equations) can be realized as a gradient flow on a manifold where the
distance is determined by the Wasserstein metric. D. Kinderlehrer and I have
exploited this idea to construct numerical schemes to approximate the solution
of such problems. This approach is interesting since it is one of the only
numerical schemes that works directly with the weak topology, and allows
the use of discontinuous basis functions for approximating solutions of (second
order) parabolic problems.
- Liquid Crystal Flows: The equations governing the motion of liquid
crystals represent a formidable system of pde's; moreover, the natural weak
problem associated for the problem can not be shown to be well posed.
In joint work with C. Liu we have developed numerical schemes for which discrete
versions of his recently discovered energy estimates hold and for which error
estimates can be established. Solutions computed with these codes clearly
exhibit the motion of defects with arise in the director fields. One
subtlety of this work is that Hermite interpolation (or a mixed method) is
required for a problem which formally does not appear to require this.
- Non-Convex Variational Problems and Young Measures: Certain phase
change problems in metallurgy give rise to non--convex variational problems
for which only generalized solutions exist (Young measures describing microscrutures).
To avoid the problems associated with direct simulation of the original ill
posed problem, I have proposed various algorithms which attempt to compute
the generalized solutions directly. To date the most promising of this
class of algorithms is the one analyzed by G. Dolzmann and myself for computing
the rank one convex hull of the bulk energy function.
- Discrete Norms and Poincare Inequalities: The analysis of many
numerical schemes naturally gives rise to estimates in discrete norms which
change with the geometry of the mesh. While it is frequently ``intuitively
clear'' that these discrete norms have continuous analogues, the presence
of mesh dependent constants makes it difficult to establish these conjectures.
By exploiting graph embedding techniques G. Miller and I have developed tools,
such as discrete Poincar\'e inequalities, which provide rigorous proofs of
such conjectures. These tools are also exploited in our research on mesh generation
- Rates of Convergence for Degenerate
Variational Problems: The convexification of certain
variational problems leads to degenerate variational principles
where the principle term of the energy vanishes on large sets.
Classical theorems of numerical analysis can
not establish strong convergence of minimizing sequences arising from
the finite element method. However, using
a result of Ekeland, R. Nicolaides and I were
able to not only establish strong convergence for a class of
problems, but also establish a rate in a suitably
- Optimal rates of convergence of the Stefan Problem: The classical
Stefan problem exhibits a discontinuity in the energy and a jump in the temperature
gradient, and this lack of regularity resulted in sub--optimal rates of convergence
using the classical tools of numerical analysis. By exploiting the tools of
convex analysis for non--smooth functions (in particular, a ``chain rule''
established with the Hahn--Banach theorem), J. Rulla and I were able to establish
optimal rates of convergence.