Talks
Doug Arnold. The periodic table of finite elements
Abstract: Finite element methodology, reinforced by deep mathematical analysis,
provides one of the most important and powerful toolsets for numerical simulation.
Over the past forty years a bewildering variety of different finite element spaces have
been invented to meet the demands of many different problems. The relationship
between these finite elements has often not been clear, and the techniques developed
to analyze them can seem like a collection of ad hoc tricks. The finite element
exterior calculus, developed over the last decade, has elucidated the requirements for
stable finite element methods for a large class of problems, clarifying and unifying
this zoo of methods, and enabling the development of new finite elements suited to
previously intractable problems. In this talk, we will discuss the big picture that
emerges, providing a sort of periodic table of finite element methods.
Luis Caffarelli. The homogenization of fronts and surfaces
Abstract: Most of the homogenization theory is develop for “bulk” quantities:
densities, flows, etc.
In this lecture I will focus on some homogenization processes that concern surfaces,
like phase transitions, minimal surfaces or propagating fronts. In particular, I will
describe in some detail geometric methods in the context of a flame front model
(this last part is joint work with Regis Monneau).
Bernardo Cockburn. Convergence for hybridizable discontinuous Galerkin methods
Abstract: We introduce the hybridizable discontinuous Galerkin methods for
second-order elliptic equations and place them in relation to already known discontinuous Galerkin methods, mixed methods and the continuous Galerkin methods.
Using this point of view, we review old and new work done on convergence for
discontinuous Galerkin methods. This is joint work with Ricardo Nochetto and
Wujun Zhang.
Ronald DeVore. Remarks on solving parametric elliptic problems
Abstract: We will discuss numerical strategies for solving parametric elliptic equations. Our emphasis will be on two points of divergence from the typical strategies
of Reduced Modeling. The one will be to allow discontinuous diffusion coefficients.
The second will be the possible incorporation of nonlinear methods into the numerical procedure. Our results are very preliminary but may serve to focus future
work.
Sandro Salsa. Free boundary problems with distributed sources: regularity results
Abstract: We describe new results on the regularity of the free boundary in
two-phase problems governed by second order elliptic equations with distributed
sources. In particular, Lipschitz or suitable ””flat”” free boundaries are smooth.
Joint works with Daniela de Silva and Fausto Ferrari.
Georgios Akrivis. Implicit-explicit multistep methods for a class of nonlinear
parabolic equations
Abstract: We consider the discretization of an initial value problem for a nonlinear
parabolic equation, in an abstract Hilbert space setting, by combinations of implicit
and explicit multistep schemes. We will discuss consistency and stability of the
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schemes, under certain conditions, and will derive optimal order error estimates.
The stability assumptions can be relaxed in the case of first and second order
schemes. The discretization of nonlinear convection-diffusion equations by implicitexplicit multistep schemes will also be briefly discussed.
Harbir Antil. A Stokes free boundary problem with surface tension effects
Abstract: We consider a Stokes free boundary problem with surface tension effects
in variational form. This model is an extension of the coupled system proposed by
P. Saavedra and L. R. Scott, where they consider a Laplace equation in the bulk
with Young-Laplace equation on the free boundary to account for surface tension.
The two main difficulties for the Stokes free boundary problem are: the vector
curvature on the interface, which causes problem to write a variational form of
the free boundary problem and the existence of solution to Stokes equations with
1+1/p′
Navier-slip boundary conditions for Wp
domains (minimal regularity). We will
demonstrate the existence of solution to Stokes equations with Navier-slip boundary
conditions using a perturbation argument for the bended half space followed by
1+1/p′
standard localization technique. The Wp
regularity of the interface allows us
to write the variational form for the entire free boundary problem, we conclude
with the well-posedness of this system using a fixed point iteration.
Eberhard Bänsch. A posteriori error estimates for approximations of the NavierStokes equations by projection schemes
Abstract: Thanks to their conceptional and computational simplicity projection
schemes are very popular and often rather efficient tools for the computational
solution of the time dependent Navier-Stokes equations.
In this talk we present a posteriori error estimates for the (continuous in space)
semi-discrete case, thus measuring the splitting error. We present estimates for the
standard pressure correction scheme with backward Euler discretization as well as
for a BDF2 scheme in so called “rotation form”.
To the best of our knowledge these are the first results regarding a posteriori control
for projection schemes.
Soeren Bartels. Finite element approximation of functions of bounded variation
Abstract: Various phenomena involving free boundaries such as damage or plasticity require the description of physical quantities with discontinuous functions.
One approach to their mathematical modeling is based on the space of functions of
bounded variation which includes functions that are discontinuous and may jump
across lower dimensional subsets. Numerical methods for their approximate solution are often based on regularizations which typically lead to restrictive conditions
on discretization parameters. We try to avoid such modifications and discuss the
convergence of discretizations with different finite element spaces, the iterative solution of the resulting finite-dimensional nonlinear systems of equations, and adaptive mesh-refinement techniques based on rigorous a posteriori error estimates for
a model problem related to image processing. The application of the techniques to
total variation flow, very singular diffusion processes, and segmentation problems
will be addressed.
Part of this talk is based on joint work with Ricardo H. Nochetto (University of
Maryland, USA) and Abner J. Salgado (University of Maryland, USA).
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Andrea Bonito. Alternative Representation of Fractional Power of Self-Adjoint
Elliptic Operators
Abstract: Taking advantage of the spectral properties of elliptic self-adjoint operators, we deduce a representation formula for their fractional powers. We show
that in this context, fractional powers reduce to a singular integral over the positive
real numbers of a perturbation of the original operator.
Then, we deduce a novel numerical algorithm for the approximation of the fractional
powers of such operators. A quadrature formula approximating the one dimensional
singular integral is proposed while standard finite element methods are advocated
for the space discretization. The particularities of the proposed method is that the
quadrature points are distributed adequately to capture the (known) singularity
and it reduces to independent elliptic solves in space. The latter implies efficient
scalability for parallel implementations.
We finally discuss optimal a-priori error estimates in terms of the number of degree of freedoms used for the space discretization and the number of quadrature
points.
Claudio Canuto. Adaptive high-order methods
Abstract: I will report on joint work with Ricardo and Marco Verani on adaptive
algorithms for Fourier or Legendre spectral methods. The nature of the approximation suggests a more aggressive attitude than for finite-order methods; on the
other hand, the complexity analysis must cope with sparsity classes in which the
best N-term approximation error decays faster than algebraically. This leads to
some surprise. In the last part of my talk, I will discuss the possibility of extending
our framework of analysis to spectral-element/h-p fem discretizations, where the
dilemma ””refine or enrich”” poses new challenges.
Zhiming Chen. An Adaptive Immersed Finite Element Method with Arbitrary
Lagrangian-Eulerian Scheme for Parabolic Equations in Variable Domains
Abstract: An adaptive immersed finite element method based on the a posteriori error estimate for solving elliptic equations with non-homogeneous boundary
condition in general Lipschitz domain is proposed. The underlying finite element
mesh need not to fit the boundary of the domain. Optimal a priori error estimate
of the proposed immersed finite element method is proved. The immersed finite
element method is then used to solve parabolic problems in time variable domains
together with an arbitrary Lagrangian-Eulerian (ALE) time discretization scheme.
An a posteriori error estimate for the fully discrete immersed finite element method
is derived which can be used to adaptively update the time step sizes and finite
element meshes at each time step. Numerical results are reported to support the
theoretical results. This is a joint work with Zedong Wu and Yuanming Xiao.
Long Chen. Multigrid methods for degenerate and singular elliptic equations
Abstract: In this talk, we will present fast multilevel methods for the approximate
solution of the discrete problems that arise from the discretization of fractional
Laplacian. The fractional Laplacian is a nonlocal operator. To localize it, we solve
a Dirichlet to a Neumann-type operator via an extension problem. However, this
comes at the expense of incorporating one more dimension to the problem, thus
motivates our study of multilevel methods. We shall use the multilevel framework
developed by Xu and Zikatanov and we show nearly uniform convergence of a
multilevel method for a class of general degenerate elliptic equations. Because
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of the singularity of the solution, anisotropic elements in the extended variable
are needed in order to obtain quasi-optimal error estimates. For this reason, we
also consider a multigrid method with a line smoother and obtain nearly uniform
convergence rates.
This is a joint work with Blanca Ayuso de Dios, Ricardo H. Nochetto, Enrique
Ot’arola and Abner J. Salgado.
Georg Dolzmann. Modelling and simulation of vectorfields on surfaces
Abstract: We introduce a nonlinear model for the evolution of biomembranes
driven by the gradient flow of a novel elasticity functional describing the interaction
of a director field on a membrane with its curvature. In the linearized setting of
a graph we present a practical finite element method (FEM), and prove a priori
estimates. We derive the relaxation dynamics for the nonlinear model on closed
surfaces and introduce a parametric FEM. We present numerical experiments which
agree well with the expected behavior in model situations. This is joint work with
Soeren Bartels, Ricardo Nochetto and Alexander Raisch
Willy Dörfler. A posteriori error estimation for indefinite Helmholtz problems
Abstract: We study the possibilities to get a posteriori error estimates for the
solution of Helmholtz problems that do not or only weakly depend on the wave
number.
Lucia Gastaldi. Finite elements for Immersed Boundary Method
Abstract: The aim of this talk is to discuss the performances of finite elements in
the space discretization of the Immersed Boundary Method. Immersed boundary
solution is characterized by pressure discontinuities at fluid structure interface.
We analyze some popular Stokes elements such as Hood-Taylor and BercovierPirennau spaces together with some lowest order stabilizations. In particular, we
investigate the local mass conservation properties of the considered schemes and
analyze new schemes with enhanced pressure approximation, which guarantee a
better local discretization of the divergence free constraint. Results show that
the enhanced pressure spaces are a significant cure for the well known “boundary
leakage” affecting IBM.
Christian Kreuzer. Design and convergence analysis for an adaptive discretization of the heat equation
Abstract: We present an adaptive fully discrete space-time finite element method
for the heat equation. The algorithm is based on a classical adaptive time-stepping
scheme supplemented by an additional control of a potential energy increase of the
discrete solution originating from coarsening of the spatial meshes. This control
allows to prove critical energy estimates in terms of given data from which one
can derive an apriori computable minimal time-step-size, which is sufficient for the
required tolerance. The minimal step-size is used by the algorithm and guarantees
that the final time is reached in finitely many time-steps and within a prescribed
tolerance.
The minimal time-step-size has also a very positive effect in simulations. We present
numerical experiments that show a significant speedup compared to classical timestepping schemes since too small time-steps are avoided.
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Irene Kyza. Error control and adaptivity for linear Schrödinger equations in the
semiclassical regime
Abstract: We derive optimal order a posteriori error bounds for a fully discrete
Crank–Nicolson finite element scheme for linear Schrödinger equations. The derivation of the estimators is based on the reconstruction technique; in particular, we
introduce a novel elliptic reconstruction that leads to estimates which reflect the
physical properties of the equation. Our analysis also includes rough potentials.
Using the obtained a posteriori error estimators, we further develop and analyze
an existing time-space adaptive algorithm, and we apply it to the one-dimensional
Schrödinger equation in the semiclassical regime. The adaptive algorithm reduces
the computational cost drastically and provides efficient error control for the solution and the observables of the problem, especially for small values of the Planck
constant.
This is a joint work with Th. Katsaounis.
Omar Lakkis. Galerkin methods for fully nonlinear elliptic equations
Abstract: Fully nonlinear elliptic equations have been long left without a proper
treatment by Galerkin methods. I will review recent advances in efficient methods
for the solution of fully nonlinear elliptic equations, such as Monge-Ampère and
Pucci equations. I will focus on the technique of Hessian-recovery and nonvariational Galerkin method.
This talk results from joint effort with Tristan Pryer (Kent, England).
Jae-Hong Pyo. Error estimates for the second order semi-discrete stabilized Gauge–Uzawa
method for the Navier–Stokes equations
Abstract: The Gauge–Uzawa method [GUM], which is a projection type algorithm
to solve the time depend Navier–Stokes equations, has been constructed in [2] and
enhanced in [3, 5] to apply to more complicated problems. Even though GUM possesses many advantages theoretically and numerically, the studies on GUM have
been limited on the first order backward Euler scheme except normal mode error
estimate in [4]. The goal of this paper is to research the 2nd order GUM. Because
the classical 2nd order GUM which is studied in [4] needs rather strong stability
condition, we modify GUM to be unconditionally stable method using BDF2 time
marching. The stabilized GUM is equivalent to the rotational form of pressure correction method and the errors are already estimated in [1] for the Stokes equations.
In this paper, we will evaluate errors of the stabilized GUM for the Navier–Stokes
equations. We also prove that the stabilized GUM is an unconditionally stable
method for the Navier–Stokes equations. So we conclude that the rotational form
of pressure correction method in [1] is also unconditionally stable scheme and that
the accuracy results in [1] are valid for the Navier–Stokes equations.
[1] J.L. Guermond and J. Shen On the error estimates of rotational pressurecorrection projec- tion methods, Math. Comp., 73 (2004), 1719-1737.
[2] R.H. Nochetto and J.-H. Pyo, A finite element Gauge-Uzawa method. Part I :
the Navier- Stokes equations, SIAM J. Numer. Anal., 43, (2005), 1043–1068.
[3] R.H. Nochetto and J.-H. Pyo, A finite element Gauge-Uzawa method. Part II :
Boussinesq Equations, Math. Models Methods Appl. Sci., 16, (2006), 1599–1626.
[4] J.-H. Pyo and J. Shen, Normal Mode Analysis of Second-order Projection Methods for In- compressible Flows, Discrete Contin. Dyn. Syst. Ser. B, 5, (2005),
817–840.
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[5] J.-H. Pyo and J. Shen, Gauge Uzawa methods for incompressible flows with
Variable Density, J. Comput. Phys., 211, (2007), 181–197.
Rodolfo Rodriguez. Numerical approximation of Beltrami fields
Abstract: Vector fields H satisfying curl H = λH, with λ being a scalar field,
are called force-free fields. This name arises from magnetohydrodynamics, since
a magnetic field of this kind induces a vanishing Lorentz force: F := J × B =
curl H × (µH). In 1958 Woltjer [W] showed that the lowest state of magnetic
energy density within a closed system is attained when λ is spatially constant. In
such a case H is called a linear force-free field and its determination is naturally
related with the spectral problem for the curl operator. The eigenfunctions of this
problem are known as free-decay fields and play an important role, for instance, in
the study of turbulence in plasma physics.
The spectral problem for the curl operator, curl H = λH, has a longstanding
tradition in mathematical physics. A large measure of the credit goes to Beltrami
[B], who seems to be the first who considered this problem in the context of fluid
dynamics and electromagnetism. This is the reason why the corresponding eigenfunctions are also called Beltrami fields. On bounded domains, the most natural
boundary condition for this problem is H · n = 0, which corresponds to a field
confined within the domain. Analytical solutions of this problem are only known
under particular symmetry assumptions. The first one was obtained in 1957 by
Chandrasekhar and Kendall [CK] in the context of astrophysical plasmas arising in
modeling of the solar crown.
More recently, some numerical methods have been introduced to compute forcefree fields in domains without symmetry assumptions [BA1,BA2]. In this work,
we propose a variational formulation for the spectral problem for the curl operator
which, after discretization, leads to a well-posed generalized eigenvalue problem.
We propose a method for its numerical solution based on Nédélec finite elements of
arbitrary order. We prove spectral convergence, optimal order error estimates and
that the method is free of spurious-modes. Finally we report some numerical experiments which confirm the theoretical results and allow us to assess the performance
of the method.
[B] E. Beltrami,Considerazioni idrodinamiche. Rend. Inst. Lombardo Acad.
Sci. Let., vol. 22, pp. 122–131, (1889). (English translation: Considerations on
hydrodynamics, Int. J. Fusion Energy, vol. 3, pp. 53–57, (1985).)
[BA1] T.Z. Boulmezaud, T. Amari,Approximation of linear force-free fields
in bounded 3-D domains. Math. Comp. Model., vol. 31, pp. 109–129, (2000).
[BA2] T.Z. Boulmezaud, T. Amari,A finite element method for computing
nonlinear force-free fields. Math. Comp. Model., vol. 34, pp. 903–920, (2001).
[CK] S. Chandrasekhar, P.C. Kendall,On force-free magnetic fields. Astrophys. J., vol. 126, pp. 457–460, (1957).
[N] J.C. Nédélec, Mixed finite elements in R3 . Numer. Math., vol. 35, pp.
315–341, (1980).
[W] L. Woltjer, A theorem on force-free magnetic fields. Prod. Natl. Acad.
Sci. USA, vol. 44, pp. 489–491, (1958)
Alfred Schmidt. FEM for phase transitions in welding processes
Abstract: We consider the simulation of solid-liquid phase transitions in the context of welding processes and similar. The model includes heat transfer, melting
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and solidification, and free surface melt flow. Special care is needed expecially
where different free boundaries meet.
The talk presents joint work with Eberhard Bänsch and Jordi Paul (Erlangen) and
Mischa Jahn and Andreas Luttmann (Bremen).
Kunibert G. Siebert. Adaptive finite elements for PDE constrained optimal control problems
Abstract: Many optimization processes in science and engineering lead to optimal control problems where the sought state is a solution of a partial differential
equation (PDE). Control and state may be subject to further constraints. The
complexity of such problems requires sophisticated techniques for an efficient numerical approximation of the true solution. One particular method are adaptive
finite element discretizations.
We report on ongoing research about control constrained optimal control problems.
We give a summary about recent findings concerning sensitivity analysis, a posteriori error control, and convergence of adaptive finite elements.
This is joint work with Fernando D. Gaspoz (Stuttgart).
Marco Verani. Hierarchical a posteriori error estimators for the mimetic discretization of elliptic problems
Abstract: We present a posteriori error estimates of hierarchical type for the
mimetic discretization of elliptic problems. Under a saturation assumption, the
global reliability and efficiency of the proposed a posteriori estimators are proved.
Several numerical experiments assess the actual performance of the local error indicators in driving adaptive mesh refinement algorithms based on different marking
strategies.
(Joint work with P.F. Antonietti, L. Beirao Da Veiga and C. Lovadina)
Shawn Walker. A new mixed formulation for a sharp interface model of Stokes
flow and moving contact lines
Abstract: Two phase fluid flows on substrates (i.e. wetting phenomena) are important in many industrial processes, such as micro-fluidics and coating flows. These
flows include additional physical effects that occur near moving (three-phase) contact lines. We present a new 2-D variational (saddle-point) formulation of a Stokesian fluid with surface tension that interacts with a rigid substrate. The model
is derived by an Onsager type principle using shape differential calculus (at the
sharp-interface, front-tracking level) and allows for moving contact lines and contact angle hysteresis through a variational inequality. We prove the well-posedness
of the time semi-discrete and fully discrete (finite element) model and discuss error
estimates. Simulation movies will be presented to illustrate the method. We conclude with some discussion of a 3-D version of the problem as well as future work
on optimal control of these types of flows.