(January 25) Dr. Ari Stern: Hilbert complexes, finite element exterior calculus, and problems on Riemannian hypersurfaces — In recent years, the success of "mixed" finite element methods has been shown to have surprising connections with differential geometry and algebraic topology---particularly with the calculus of exterior differential forms, de Rham cohomology, and Hodge theory. In this talk, I will discuss how the notion of "Hilbert complex," rather than "Hilbert space," provides the appropriate functional-analytic setting for the numerical analysis of these methods. Furthermore, I will present some recent results that extend this analysis from polyhedral regions in Euclidean space to arbitrary Riemannian manifolds. As a direct consequence, our analysis also generalizes several key results on "surface finite element methods" for the approximation of elliptic PDEs on hypersurfaces (e.g., membranes undergoing geometric evolution).

(February 1) Prof. Elisabeth Ullman: Efficient Iterative Solvers for Stochastic Galerkin Discretizations of the Lognormal Diffusion Problem — We consider stochastic Galerkin discretizations of a steady-state diffusion problem with stochastically nonlinear, lognormally distributed diffusion coefficient. The iterative solution of the resulting large, coupled system of Galerkin equations is computationally demanding, since matrix-vector products with the Galerkin matrix are expensive and the Galerkin matrix is ill-conditioned with respect to the statistical parameters of the problem. We consider a reformulated version of the lognormal diffusion problem in terms of a stochastic convection-diffusion problem with random convective velocity which depends linearly on a fixed number of statistically independent Gaussian random variables. The associated Galerkin matrix is nonsymmetric but sparse and allows for fast matrix-vector multiplications with optimal complexity. We construct and analyze two block diagonal preconditioners for this Galerkin matrix for use with the generalized minimal residual method (GMRES). We test the efficiency of the proposed preconditioning approaches and compare the iterative solver performance for a lognormal diffusion problem posed as convection-diffusion and diffusion problem.

(February 8) Dr. Markus Schmuck: A porous media approximation of the Stokes-Poisson-Nernst-Plank equations — We first introduce a classical continuum model which allows to describe essential electrokinetic phenomena as electro-phoresis and -osmosis. Applications and corresponding theory range from design of microfluidic devices, energy storage devices, semiconductors over modeling communication in biological cells by nanopores. Based on this classical formulation, we derive effective macroscopic equations which describe binary symmetric electrolytes in porous media. The inhomogeneous structure of porous materials naturally induces additional nonlinearities which are called "material tensors". A better understanding of geometric effects in heterogeneous media on ionic transport is expected by the new formulation. The numerical advantage bases on a strong reduction of the degrees of freedom by upscaling the microstructure. The results are gained by the two-scale convergence method. In the remaining part of the talk, we motivate the derivation of first error bounds between the exact microscopic solution and its upscaled macroscopic approximation.

(February 10) Prof. Peter Constantin: Regularity issues for the Surface Quasi-Geostrophic and related equations — "The Surface Quasi-Geostrophic Equation" is a nonlinear PDE which has a geophysical origin, but is mainly used as a theoretical testing ground for ideas about singularities in incompressible fluids. I'll motivate the model and present recent results, some numerical, mostly analytical. The main problem is that of global regularity for the inviscid equation, and that is still open.

(February 11) Prof. Peter Constantin: Complex Fluids — The talk will be about some of the models used to describe fluids with iparticulate matter suspended in them. Some of these models are very complicated. After a bit of history and a review of known results, I will try to point out some open problems, isolate some of the mathematical difficulties, and illustrate some of the phenomena on simpler didactic models.

(February 15) Prof. A. Javier Sayas: Energy estimates in semidiscrete time-domain boundary integral equations — Boundary integral operators in the time domain offer competitive ways to solve exterior scattering problems for different kinds of transient (acoustic, elastic, visco-elastic,?electromagnetic,...) waves. They also provide exact absorbing boundary conditions that can be placed arbitrarily close to the support of source terms or inhomogeneities. In this talk I will address the problem of how space discretization with Galerkin BEM redistributes the energy, leaking energy to the interior of the scatterer. In the case of transmission problems, BEM-FEM space discretization creates a ghost wave in the interior domain. This effect can be shown by considering abstract wave equations with exotic transmission condition that are exactly satisfied by the semidiscrete equations. This novel energy analysis can be used to better understand the correct balance of energy of Boundary Integral Methods and might lead to evidence for or against the possibility of using non-symmetric BEM-FEM coupled methods in the time domain. Because of its generality, the conclusions are valid both for Galerkin or Convolution Quadrature discretizations in time.

(February 22) Prof. Petr Plechak: Born-Oppenheimer approximation and accuracy of molecular dynamics — I shall discuss recent results from the joint work with C. Bayer, H. Hoel, A. Szepessy and R. Tempone concerning accurate approximation of Schroedinger observables in molecular systems with an electron spectral gap. We present error estimates that can be used for estimating accuracy of molecular dynamics as an approximation of the evolution of heavy nuclei in a many-body quantum system. The presented approach is based on the study of the time-independent Schroedinger equation and thus differs from a more standard analysis derived from time-dependent Schroedinger equation. It gives a different perspective on the Born-Oppenheimer approximation, Schroedinger Hamiltonian systems and numerical simulations in molecular dynamics at micro-canonical ensemble.

(March 1) Prof. Adrian Lew: Discontinuous Galerkin Methods in Solid Mechanics — Some problems in Solid and Structural mechanics require special care when analyzed via finite element approximations. Typical examples are problems that involve kinematic constraints, such as in incompressible elasticity or Reissner-Mindlin plate models, and problems involving moving boundaries, such as evolving cracks, phase transition interfaces or shape optimizations. Nonlinearities in the material behavior only exacerbate these difficulties, and immediately rule out many of the proposed solutions.

In this talk I will show how under these circumstances Discontinuous Galerkin methods provide an attractive and advantageous alternative. The overarching idea I will convey is that by relaxing the constraint of having continuous displacements across element boundaries, Discontinuous Galerkin methods are able to impose other kinematic constraints in the problem and still provide accurate solutions. I will demonstrate it by showcasing the performance of a class of Discontinuous Galerkin methods we introduced in a variety of circumstances. First, in nonlinear elasticity problems involving different kinematic constraints. Second, in a class of immersed boundary methods, which sidestep the need for automatic remeshing in problems with evolving boundaries by embedding the boundary in any mesh. And finally, in the accurate solution of the stress and displacement fields around cracks when cracks are "embedded" in the mesh, as in extended finite element methods. In the three cases, I will comment on recent convergence results we obtained. In all cases, a special interpolant taking advantage of the discontinuities in the finite element space had to be constructed, since standard interpolants in conforming finite element spaces can be guaranteed to converge only at suboptimal rates. I will conclude the talk by briefly commenting on an adaptive stabilization technique we created to enhance the robustness of the method in highly nonlinear problems. Some of these ideas carry over to a wider variety of popular methods in Solid Mechanics, such as Enhanced Strain Methods.

(March 8) Prof. Christian Klingenberg: A numerical scheme for ideal MHD and its application to turbulence simulations in astrophysics — The reported work gives an overview of a code for ideal MHD developed at the University of Oslo applied to astrophysical simulations done at Wrzburg University both of which I have contributed to. The main ingredients of the code are: finite volume scheme, an appoximate Riemann solver based on relaxation, using a Powell source term, second order positivity preserving reconstructions. The application consist in doing driven turbulence simulations in order to study their influence on star formation.

(March 15) Prof. Alina Chertock: Sinking, merging and stationary plumes in a coupled chemotaxis-fluid model: a high-resolution numerical approach — Aquatic bacteria like Bacillus subtilis are heavier than water yet they are able to swim upwards an oxygen gradient and concentrate in a layer below the water surface, which will undergo Rayleigh-Taylor type instabilities for sufficiently high concentrations. Recently, a simplified chemotaxis-fluid system has been proposed as a model for modestly diluted cell-suspensions. It couples a convective chemotaxis system for the oxygen-consuming and oxyotactic bacteria with the incompressible Navier-Stokes equations subject to a gravitational force proportional to the relative surplus of the cell-density compared to the water-density.

In this talk, I will present a high-resolution vorticity-based hybrid finite-volume finite-difference scheme, which allows to investigate the nonlinear dynamics of a two-dimensional chemotaxis fluid system. In particular, I will show selected numerical examples, which illustrate (i) the formation of sinking plumes, (ii) the possible merging of neighboring plumes and (iii) the convergence towards numerically stable stationary plumes. The examples with stable stationary plumes show how the surface-directed oxyotaxis feeds continuously cells into a high-concentration layer near the surface, from where the fluid-flow (recurring upwards in the space between the plumes) transports the cells into the plumes, where then gravity makes the cells sink and constitutes the driving force in maintaining the fluid-convection and, thus, in shaping the plumes into (numerically) stable stationary states.

Joint work with K. Fellner, A. Kurganov, A. Lorz and P. Markowich.

(March 29) Prof. Georg Dolzmann: Modeling and simulation of vectorfields on membranes — The fundamental models for lipid bilayers are curvature based and neglect the internal structure of the lipid layers. In this talk, we explore models with an additional order parameter which describes the orientation of the lipid molecules in the membrane and compare their predictions based on numerical simulations. This is joint work with Soeren Bartels (Bonn) and Ricardo Nochetto (College Park).

(April 5) Prof. Jay Gopalachisnan: Design and analysis of the new Discontinuous Petrov-Galerkin schemes — Starting with simple examples we will show how to design numerical methods of the DPG (Discontinuous Petrov-Galerkin) type. Petrov-Galerkin method find approximations in a trial space by weakly imposing the equations using a test space. Our design principle is that while trial spaces must have good approximation properties, the test space must be chosen for stability. When this idea is applied to discontinuous finite element spaces, we obtain methods that exhibit remarkable stability properties with respect to mesh size and approximating polynomial degrees.

(April 12) Prof. K.G. van der Zee: Goal-Oriented Error Estimation and Adaptivity of Interface Problems: Free Boundaries and Diffuse Interfaces — Problems with evolving interfaces arise in many applications in science and engineering. To obtain numerical approximations various approaches have been developed, for example moving-domain (ALE) methods and diffuse-interface models. Interface problems are inherently nonlinear, displaying multiple length and time scales, and therefore would benefit highly from error estimation and adaptive techniques.

In this talk, we consider goal-oriented error estimation and adaptivity of interface problems. Goal-oriented error estimates are dual-based a posteriori estimates of the error in output functionals of the solution (the quantities of interest). The estimates rely on the computation of a dual (linearized-adjoint) problem, which can be highly nontrivial for interface problems. We consider two examples: a free-boundary problem and a time-dependent diffuse-interface model.

For free-boundary problems, such as fluid-structure interaction, the linearization of the geometric nonlinearity required for the dual problem poses a significant complication. We show that the linearization using shape derivatives yields a dual problem corresponding to a curvature-dependent boundary-value problem. For diffuse-interface problems, such as the Cahn-Hilliard equation of phase separation, the linearized-adjoint problem is a backwards-in-time evolution equation with unstable growth in the phase-separating regime. For both problems we present numerical experiments that show the effectivity of the dual-based error estimates.

(April 19) Dr. Irene Kyza: Error analysis of time-discrete higher order ALE formulations — ALE formulations are useful when approximating solutions of problems in deformable domains, such as fluid-structure interactions. For realistic simulations involving fluids in 3d, it is important that the ALE method is at least of second order of accuracy. Second order ALE methods in time without any constraint on the time step do not exist so far in the literature and the role of the so called geometric conservation law (GCL) for stability and accuracy is not clear. We propose discontinuous Galerkin (dG) methods of any order in time for a time dependent advection-diffusion model problem in moving domains. We prove that our proposed numerical schemes are unconditionally stable and that the conservative and non conservative formulations are equivalent, when the involved integrals are computed exactly. Both a priori and a posteriori error analyses are provided. The final error estimates are of optimal order of accuracy. When using quadrature for the evaluation of the integrals, we give a sufficient condition for preserving the stability and the accuracy of the numerical schemes, without any constraint on the time-step. This condition is a generalization of the GCL. Stability and optimal order error estimates, under a mild restriction on the time step, are also obtained when applying the natural for the dG method in time Radau quadrature. Numerical examples confirm our theoretical results.

This is joint work with Andrea Bonito and Ricardo H. Nochetto

(April 26) Dr. Josef Sifuentes: The stability of GMRES convergence, with application to preconditioning by approximate deflation — How does GMRES convergence change when the coefficient matrix is perturbed? Through resolvent estimates we develop simple, general bounds to quantify the lag in convergence such a perturbation can induce. This analysis is particularly relevant for preconditioned systems, where an ideal preconditioner is only approximately applied in practical computations. To illustrate the utility of this theory, we combine our analysis with Stewart's invariant subspace perturbation theory to develop rigorous bounds on the performance of approximate deflation preconditioning using Ritz vectors.

(May 3) Prof. Florian Potra: Interior-Point Methods — Interior point methods have revolutionized the field of mathematical programming over the past three decades. They have been used for proving polynomial complexity for different classes of mathematical programming problems, and they have been implemented in very efficient software packages for solving large scale optimization problems arising in a variety of applications. While the implemented interior point methods may not always have proven computational complexity, they typically posses superlinear convergence. The talk highlights the most relevant results on the polynomial complexity and superlinear convergence of interior point methods in the literature, and presents some recent results obtained by the speaker and his collaborators.

(May 10) Prof. Bernardo Cockburn: Superconvergent DG methods for elliptic problems — We propose a new error analysis of a large class of finite element methods for second order elliptic problems which includes the hybridized version of the main mixed methods and the hybridizable discontinuous Galerkin methods.

The main feature of this approach is that it reduces the main difficulty of the analysis to the verification of some properties of an auxiliary, locally defined projection and of the local spaces defining the methods. They guarantee the optimal convergence of the approximate flux and the superconvergence of an element-by-element postprocessing of the scalar variable.

New mixed and hybridizable discontinuous Galerkin methods with these properties are devised which are defined on squares, cubes and prisms.

(May 17) Prof. Yoichiro Mori: A Three-Dimensional Model of Electrodiffusion and Osmosis — We introduce a three-dimensional model of cellular electrical activity. This model takes into account the three-dimensional geometry of biological tissue as well as ionic concentration dynamics, both of which are neglected in conventional models of electrophysiology. This model is applied to study an anomalous mode of cardiac action potential propagation: cardiac propagation without gap junctions. Then, we shall discuss an extension of this model to incorporate osmotic and membrane mechanical effects. The salient feature of this extension is that it satisfies a natural free energy identity. We discuss how this free energy identity has led to the resolution of a longstanding question on the stability of steady states of pump-leak models of cell volume control.

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