|
PDE Seminar, 2006
Next seminar:
The seminars aim to share knowledge of the results achieved and open discussion on the ongoing research. The style is informal and the speakers include both students and researchers at CMA and visiting fellows. Student at master level are welcome to take active part in the series.
Time and place
Thursday 10.15-1100 in room B1036.
Preliminary list of speakers:
Nov 9. Christopher Dyken and Michael Floater (CMA)
Title: Web-splines for solving PDE's
Abstract: We will give an informal introduction to the use of web-splines (Weighted Extended B-splines) for solving elliptic PDE's via a finite element method, due to Hollig, Reif, and Wipper. The method requires a "weight" function that behaves like the signed distance to the boundary of the domain but is preferably smooth. We describe a new approach to the latter, based on mean value interpolation.
Nov 2 . Darko Mitrovic (NTNU)
See the CMA Guest Lectures
Oct 26: Kenneth H. Karlsen (CMA)
Title: Mathematical and numerical aspects of some integrable equations
Abstract: The Korteweg-de Vries (KdV) equation models weakly nonlinear unidirectional long waves, and arises in various physical contexts. For example, it models surface waves of small amplitude and long wavelength on shallow water. The Camassa-Holm equation entered the arena in the early 1990s. In one interpretation, it models the propagation of unidirectional shallow water waves on a flat bottom. The Camassa-Holm equation is a water wave equation of quadratic order in an asymptotic expansion for unidirectional shallow water waves described by the incompressible Euler equations, while the KdV equation appears as first order in this expansion. In another interpretation it describes finite length, small amplitude radial deformation waves in cylindrical compressible hyperelastic rods, and in this context the equation is often referred to as the hyperelastic-rod wave equation. The Camassa- Holm equation possesses many interesting properties, like its bi- Hamiltonian structure and complete integrability. Moreover, it has an infinite number of non smooth solitary wave solutions called peakons. Related equations are the Hunter-Saxton and Degasperis-Procesi equations, both of which have forms similar to the Camassa Holm equation, where the former also shows up in simple models for liquid crystals.
In fact there is a whole family of related equations, most of them remain mathematically and numerically unexplored. In this talk I will review recent work on the well-posedness of generalized hyperelastic-rod wave equations and the Degasperis- Procesi equation. Additionally, I will discuss a recent result showing that as the diffusion and dispersion parameters tend to zero in Camassa-Holm type equations containing nonlinear dispersive effects as well as fourth order dissipative effects, with a condition on the relative balance between these two parameters, smooth solutions of the Camassa-Holm type equation converge to discontinuous weak solutions of a scalar conservation law (the Burgers equation). Finally, I will review recent work on judiciously chosen finite difference schemes for the Camassa-Holm, Hunter-Saxton, and Degasperis- Procesi equations, for which the convergence to the exact solutions can be rigorously established.
This talk is based on joint work with G. M. Coclite (Bari), H. Holden (Trondheim), N. H. Risebro (Oslo).
Oct 19: Boris Andreianov (Universite de Franche-Comte, Besancon, France
)
Title: On strong traces and boundary conditions for hyperbolic conservation laws
Abstract: (joint work with Karima Sbihi (Strasbourg) )
Since the classical work of C.Bardos, A.-Y. LeRoux and C.Nedelec, it is well know how to interprete the (formal) Dirichlet boundary conditions for solutions of scalar conservation laws admitting pointwise traces on the boundary.
It has recently been realized (G.-Q.Chen-M.Rascle, A.Vasseur, and E.Yu.Panov) that such pointwise traces do exist in a fairly general situation, for entropy solutions which are only supposed to be bounded (while the BV regularity was required for the Bardos-LeRoux-Nedelec result).
Although weak trace techniques by F.Otto are also available nowadays, it seems quite simpler to use the pointwise caracterization of the ''effective'' boundary conditions induced by the formal Dirichlet conditions. This is also true for other ''dissipative'' boundary conditions such as the zero- flux condition, recently investigated by R.Burger, H.Frid and Kenneth Karlsen. In fact, in their work the flux function was taken is such a form that the ''effective'' boundary condition coincides with the formal one.
The goal of this talk is to suggest what should be the right ''effective'' boundary condition formulation for a general dissipative boundary condition ( condition expressed by the requirement that the couple ( u , f(u).n ) belongs to a given maximal monotone graph \beta ; here u stands for the solution on the boundary, f(.) is the flux function and n is the exterior unit normal vector to the boundary of our domain ).
We give an appropriate up-to-the boundary entropy formulation, taking into account the existence of strong traces. We show uniqueness and L 1 contraction results. We also suggest a general scheme for proving convergence of sequences of approximating solutions possessing nice compactness properties. Existence follows in some cases : one can first make plenty of simplifying assumptions and then get rid of some of them, by consecutive well-choosen approximation procedures.
Oct 12: Trygve Karper (CMA)
Title: Convergence of the Discontinuous Galerkin Method for Discontinuous Solutions
Abstract: The lecture will be a presentation of the paper ''Convergence of the Discontinuous Galerkin Method for
Discontinuous Solutions'' by Noel J. Walkington. In this paper Walkington manages to prove convergence of a Discontinuous Galerkin scheme for a class of scalar conservation laws typically representing mass conservation in various flow problems.
The abstract given in the paper by Noel J. Walkington:
We consider linear first order scalar equations of the form p_t + Div (pv) + ap = f with appropriate initial and boundary conditions. It is shown that approximate solutions using
the discontinuous Galerkin method (space and time) will converge in L 2 [0,T,L 2 (Omega)] when the coefficients v and a and data f satisfy the minimal assumptions required to establish existence and uniqueness of solutions. In particular, v need not be Lipschitz, so characteristics of the equation may not be defined, and the solutions being approximated may not have bounded variation.
Sep 21: Franz Georg Fuchs (CMA)
Title: Euler's Elastica and Curvature-Based Inpainting
Abstract: Image inpainting is a special image restoration problem, where a missing part of the image has to be recovered. The advantage of the elastica inpainting model is, that it completes broken isophotes, i.e. level lines, very ''roundly'' in the inpainting domain - in contrast to most other existing inpainting schemes, for instance the TV-inpainting based on minimizing the BV-norm. A straight completion of edges leads to "unnatural" kinks on the boundary of the inpainting domain.
A curve is said to be Euler's elastica if it is the equilibrium curve of the elasticity energy. Euler obtained the energy in studying the steady shape of a thin and torsion-free rod under external forces in 1744. After functionalization and with the help of the level-set framework this energy is directly applicable to the image function. The resulting curvature-based energy is minimized with the gradient descent time marching by means of the Euler-Lagrange equation. This leads to a highly non-linear fourth order parabolic PDE.
A new theoretical understanding of this operator on the PDE level is given. Any explicit finite difference scheme will suffer of a severe restriction of the stepsize resulting in a very slow convergence to the steady state solution. As a result of the new interpretation one can point out how an implicit scheme allows larger stepsizes and therefore a way to speed up the convergence to the steady state is given.
Organizers:
Nils Henrik Risebro, CMA
|
