Research Summary

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 algorithms.
• 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 weighted norm.
• 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.