# EPSRC Symposium Workshop on New directions in computational partial differential equations

## Workshop on New directions in computational PDEs

#### Monday 12 - Friday 16 January 2009

#### Organisers: John Barrett (Imperial College), Charlie Elliott (Warwick), Chris Schwab (E.T.H.), Endre Süli (Oxford)

**Klaus Deckelnick** (Magdeburg) *Approximation of axisymmetric solutions of Willmore flow under Dirichlet boundary conditions*

We consider the Willmore flow of axially symmetric surfaces subject to Dirichlet boundary conditions. The corresponding evolution is described by a nonlinear parabolic PDE of fourth order for the radial variable. A suitable weak form of the equation, which is based on the first variation of the Willmore energy, leads to a semidiscrete scheme, in which we employ C1 finite elements for the approximation in space. Our main results are optimal error bounds in Sobolev norms for the solution and its time derivative. This is joint work with Friedhelm Schieweck (Magdeburg).

**S. Ganesan** (Imperial) *An accurate finite element solution of interface flows with surfactants*** ** pdf of abstract

**Ivan Graham** (Bath) *Multiscale finite elements for high-contrast elliptic interface problems*

We introduce a new multiscale finite element method which is able to accurately capture solutions of elliptic interface problems with high contrast coefficients by using only coarse quasiuniform meshes, and without resolving the interfaces. A typical application would be the modelling of flow in a porous medium containing a number of inclusions of relatively low permeability, embedded in a matrix of relatively high permeability. Our method is H1- conforming, with degrees of freedom at the nodes of a triangular mesh and requires the solution of subgrid problems for the basis functions on elements which straddle the coefficient interface, but uses standard linear approximation otherwise. A key point is the introduction of novel coefficient-dependent boundary conditions for the subgrid problems. Under moderate assumptions, we prove that our methods have (optimal) convergence rate of O(h) in the energy norm and O(h2) in the L2 norm where h is the (coarse) mesh diameter and the hidden constants in these estimates are independent of the coefficient “contrast”. The proof does not depend on perodicity or any homogenisation argument. This is joint work with Jay Chu and Tom Hou of Caltech.

**Michael Hintermueller** (Berlin) *Recent advances in optimal control of variational inequalities*

From an optimization theoretic point of view optimal control problems for variational inequalities belong to the class of mathematical programs with equilibrium constraints (MPECs, for short) in function space. These problems typically lack constraint qualifications for proving existence of Lagrange multipliers in first order characterizations. In this talk, new first order concepts based one relaxation techniques for the original problem are presented. These approaches are constructive and allow to pattern solution algorithms after the proof steps. In addition, these techniques my be intertwined with multigrid concepts. Corresponding algorithms including their convergence analysis are discussed and numerical results are presented.

**M. Hinze** (Hamburg) Optimization with PDEs in the presence of constraints – tailored discrete concepts and error analysis

pdf of abstract

**Karl Kunish** (Graz) *Semi-smooth Newton methods for optimal control of variational inequalities*

Semi-smooth Newton methods are superlinearly convergent iterative methods for non-differentiable optimization methods in function space. In the context of optimal control of variational inequalities proper regularization is required to profit from this property. Asymptotic as well as qualitative properties of this regularization are analysed.

**Angela Kunoth** (Paderborn) *Space-time adaptive wavelet methods for control problems constrained by parbolic PDEs *

Optimization problems constrained by partial differential equations (PDEs) are particularly challenging from a computational point of view: the first order necessary conditions for optimality lead to a coupled system of PDEs. For these, adaptive methods which aim at distributing the available degrees of freedom in an a¬posteriori-fashion to capture singularities in the data or domain appear to be most promising. For control problems constrained by a parabolic PDE, one needs to solve a system of PDEs coupled globally in time. For such problems, an adaptive method based on wavelets is proposed. It builds on a recent paper by Schwab and Stevenson where a single linear parabolic evolution problem is formulated in a weak space-time form and where an adaptive wavelet method is designed for which optimal convergence rates can be shown.

**Omar Lakkis **(Sussex) *Error control via elliptic reconstruction in some evolution equations*

I will review the elliptic reconstruction technique (ERT) in a posteriori error analysis and its impact on error contro and adaptivity for fully discrete schemes for parabolic equations. The flexibility of the ERT, in contrast with more standard approaches, allows a almost indiscriminate combination of various parabolic PDE techniques such as energy methods, duality methods and heat-kernel estimates. The ERT simplifies and allows interesting extensions of previous methods (Lakkis & Makridakis, 2006; Makridakis & Nochetto 2003). In particular, it provides previously unavailable error bounds for Fully Discrete Schemes, such as pointwise norm error bounds for the heat equation (Demlow, Lakkis & Makridakis, 2009) and optimal-order and to derive estimates for fully-discrete parabolic schemes using elliptic gradient-recovery estimators (Lakkis & Pryer, 2009) and for certain non-conforming methods such as spatial DGFEM (Georgoulis & Lakkis, 2009).

**John Lowengrub** (Irvine) *Multiscale models of solid tumor growth and angiogenesis*

We present and investigate models for solid tumor growth that incorporate features of the tumor microenvironment including tumor-induced angiogenesis. Using analysis and nonlinear numerical simulations, we explore the effects of the interaction between the genetic characteristics of the tumor and the tumor microenvironment on the resulting tumor progression and morphology. We account for variable cell-cell/cell-matrix adhesion in response to microenvironmental conditions (e.g. hypoxia) and to the presence of multiple tumor cell species. We focus on glioblastoma and quantify the interdependence of the tumor mass on the microenvironment and on the cellular phenotypes. The model provides resolution at various tissue physical scales, including the microvasculature, and quantifies functional links of molecular factors to phenotype that for the most part can only be tentatively established through laboratory or clinical observation. This allows observable properties of a tumor (e.g. morphology) to be used to both understand the underlying cellular physiology and to predict subsequent growth or treatment outcome, thereby providing a bridge between observable, morphologic properties of the tumor and its prognosis.

**Mitchell Luskin** (Minnesota) *Mathematical foundations for predictive and efficient quasicontinuum methods *

The development of predictive and efficient atomistic-to-continuum computational methods requires both an analysis of the error and efficiency of its many components (coupling method, model and mesh adaptivity, solution methods) as well as its integration into an efficient code capable of solving problems of technological interest. There are many choices available for the interaction between the representative atoms of the quasicontinuum method, especially between those in the atomistic and continuum regions, which has led to the development of a variety of quasicontinuum approximations. We will present criteria for determining a good choice of quasicontinuum approximation that considers trade-offs between accuracy and algorithmic efficiency. Our criteria are based on the effect of the coupling error on the goal of the computation, on the integration of the quasicontinuum approximation with model and mesh adaptivity, and on the development of efficient iterative solution methods. Joint Work with Marcel Arndt, Matthew Dobson, and Christoph Ortner

**Christoph Ortner** (Oxford) * Analysis of Quasicontinuum Methods*

I this talk, I will review some of the fundamental results in the analysis of the QC method, with particular focus on the nonlinear and non-convex nature of the problem. I will present some technical aspects for a simple next-nearest neighbour chain, however, I will comment on where the methods break down and explain some particularly interesting challenges for future research.

**A Prohl** (Tübingen) *Fabrication of aluminium – modeling, analysis, and numerics*

We consider the density-dependent magneto-hydrodynamics equations, which couples the incompressible Navier-Stokes equation with variable density and viscosity with Maxwell's equation to describe a viscous, incompressible, and electrically conducting multi-fluid. In the main part of the talk, we discuss problems to overcome to construct a convergent implicit stabilized finite element discretization: The proposed scheme satisfies a discrete energy law, and a discrete maximum principle for the positive density. These properties, together with a discrete version of the compactness result by R. DiPerna and P.L. Lions then establishes solvability, and convergence of the finite element solutions to weak solutions of the limiting problem for vanishing discretization parameters. Computational studies are provided. This is joint work with L. Banas (HW Edinburgh, UK).

**Martin Rumph** (Bonn) *Natural discretization of gradient flows - Applications to viscous thin films and Willmore flow*

The talk will focus on the natural time discretization of gradient flows based on a balance of dissipation and energy decay. Typically the dissipation is formulated in terms of a flow or transport field, whereas the energy primarily depends on a deduced quantity. This leads to a nested structure of the resulting variational problem and concepts from PDE constraint optimization come into play. Applications will include thin film flow in coating layers, the spreading of thin films on curved surfaces, and the evolution of curves and surfaces under Willmore flow.

**Carola-Bibuabe Schoenlieb** (Cambridge) *Fourth-order PDEs for image restoration *

In this talk I will present the method of PDEs, i.e., functional minimization, used in a wide range of image processing tasks, such as image denoising, deblurring, and image interpolation. In particular I am interested in nonlinear PDEs of fourth differential order appearing in image inpainting, i.e., image restoration. Thereby inpainting is the process of filling in missing parts of damaged images based on the information obtained from the surrounding areas. Digital image restoration is an important challenge in our modern computerized society: From the reconstruction of crucial information in satellite images of our earth, restoration of CT- or PET images in molecular imaging to the renovation of digital photographs and ancient artwork, digital image restoration is ubiquitous. Motivated by these applications, I investigate certain PDEs used for these tasks. We shall discuss both some of their analytic properties, the efficient numerical solution of these equations as well as the concrete real world applications (like the restoration of ancient Viennese frescoes).

**B. Stinner** (Warwick) *Elastic biomembranes involving lipid separation*

The lipids of biomembranes may separate into coexisting phases. In addition to its elastic bending energy the membrane energy then involves a line energy arising from the phase interfaces. In biophysics, equilibrium shapes are of interest, in particular with respect to budding phenomena and vesicle fission. The goal has been to numerically study energy minima by relaxing suitable initial shapes according to an appropriate gradient flow dynamics. The intermembrane domains are described using the phase field methodology leading to a pde on the membrane which is coupled to a geometric evolution law for the membrane. The discretisation is based on representing the membrane by a triangulated surface on which linear parametric FEs are defined. The convergence as the interface thickness tends to zero has been numerically analysed, and the influence of various physical parameters numerically investigated. Adaptive refinement will be briefly discussed.

**Anders Szepessy** (Stockholm) *Stochastic molecular dynamics derived from the time-independent Schrödinger Equation*

Smoluchowski, Langevin and Ehrenfest dynamics are shown to be accurate approximations of time-independent Schrödinger observables for a molecular system, in the limit of large ratio of nuclei and electron masses, without assuming that the nuclei are localized to vanishing domains. The derivation, based on characteristics for the Schrödinger equation, bypasses the usual separation of nuclei and electron wave functions and gives a different perspective on computation of observables and stochastic electron equilibrium states in molecular dynamics simulations

**E. Zuazua** (Madrid) *Dispersive methods for linear and nonlinear Schrödinger equations*

pdf of talk

See also:

Mathematics Research Centre

Mathematical Interdisciplinary Research at Warwick (MIR@W)

Past Events

Past Symposia

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