Adaptivity in space and time
Special session at the
Joint International Meeting UMI-DMV
Perugia
June 18-22, 2007
Adaptivity in space and timeSpecial session at the
Joint International Meeting UMI-DMV
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Adaptive techniques often improve the efficiency of numerical methods and thus lead to a better exploitation of the given computational resources. For certain problems, the use of adaptivity is even indispensable for the practical solvability.
In the numerical solution of ordinary differential equations (ODEs), adaptive techniques are used since the 1960s. There are various proposals for computable quantities estimating the local truncation error and their use to adapt the local time stepsize or local order of the method. Although there is a well-established heuristic ground and a great computational evidence of the efficiency of the ensuing methods, their theoretical underpinning is rather weak: e.g., there are no convergence or complexity results for the adaptive strategies used in practical codes. Renewed interest in adaptive techniques for ODEs has recently come from the need of structure preservation (e.g., reversibility) in long-term simulations of Hamiltonian systems.
In the numerical solution of partial differential equations (PDEs), adaptive finite element methods have been introduced in the late 1970s. For stationary problems, a rather elaborated theory for the derivation of a posteriori error estimators is now available. Moreover, there has been recent progress in the theoretical understanding of the convergence and complexity of the corresponding adaptive methods.
For time-dependent PDEs which can be written as abstract ODEs, the derivation of suitable computable quantities is less settled, the available algorithms for simultaneous adaptation in time and space are not so robust in practice, and questions about convergence and complexity are open.
The goal of this session is to bring together experts and researchers in space or/and time adaptivity in order to foster exchange between the fields as well as the development of their intersection.
14:30-15:25 Ricardo H. Nochetto (University of Maryland, USA)
Time reconstruction and a posteriori error
analysis for Galerkin and Runge-Kutta collocation methods
15:30-15:55 Marino Zennaro (Università di Trieste)
The potential for stability of exponential
Runge-Kutta methods for semi-linear problems
16:00-16:25 Nicola Guglielmi (Università di L'Aquila)
Multiple scales in the dynamics of forward-backward
parabolic equations
16:30-16:55 Achim Schädle (Zuse Institut Berlin)
Adaptive, fast and oblivious convolution
quadrature
17:30-17:55 Jens Lang (Technische Universität Darmstadt)
On global error estimation and control for ODEs and
parabolic PDEs
18:00-18:25 Alfred Schmidt (Universität Bremen)
Adaptive FEM for a phase transition problem with free
surface flow
18:30-18:55 Michael Schmich (Universität Heidelberg)
Adaptive space-time finite element methods for
parabolic equations
19:00-19:25 Luigi Brugnano (Università degli Studi di
Firenze)
Mesh selection and conditioning of
problems
09:00-09:25 Willy Dörfler (Universität Karlsruhe)
Adaptive finite element methods - saturation and
convergence
09:30-09:55 Giancarlo Sangalli (Università di Pavia)
Robust a-posteriori analysis for advection-dominated
problems
10:00-10:25 Stefano Berrone (Politecnico di Torino)
Adaptive discretization of parabolic equations with
discontinuous coefficients
10:30-10:55 Kunibert G. Siebert (Universität Augsburg)
Convergence analysis of adaptive finite elements for
linear parabolic problems
Adaptive discretization of parabolic equations
with discontinuous coefficients
Stefano Berrone
(Politecnico di Torino)
In many practical applications, a heat conduction problem
involving a non homogeneous medium, has to be solved. To deal with
these situations, we propose for parabolic equations with piecewise
constant coefficients robust a posteriori error estimates based on a
discretization by conforming finite elements and a classical
A-stable theta-scheme.
Our estimators are based on equation residuals. In order to derive
different estimators for the space-discretization error and the
time-discretization error, we consider a full discretization
approach instead of considering a semidiscrete formulation. These
estimates allow us to perform a control of the space-discretization
used in each time slab and of the time-step length to ensure the
error of the full discretization bounded from above and from below
in each time slab.
Some issues about the mesh compatibility conditions involved in the
coarsening step and a local timestepping are discussed. A simple
adaptive strategy is proposed and applied to some practical
situations.
Mesh selection and conditioning of
problems
Luigi Brugnano (Università degli
Studi di Firenze)
The conditioning properties of problems always assume a central role in Numerical Analysis, since they measure the amplification of perturbations on the solutions. When speaking about ODE problems, in the past years the existing connections between conditioning and stiffness of the problems have been made clear [1,2,3], allowing to obtain a new definition of "stiffness" of a problem which naturally encompasses both the case of initial and boundary value ODE problems. In addition, this has allowed to derive a new mesh selection strategy aimed to reproduce, in the discrete problem generated by the application of a given discrete method, the conditioning properties of the continuous problem. Such an approach has been recently implemented in two computational codes for ODE-BVPs, which result to be very efficient when solving stiff problems. In this talk, the main facts concerning this topic are reviewed, along with some recent advances and extensions.
Adaptive finite element methods -
saturation and convergence
Willy Dörfler
(Universität Karlsruhe)
Changing the stepsize in a computational method is a necessary tool to gain computational power. This must be based on estimates for the actual error. On to coarse grids, however, this might be dangerous, since the error can be severely underestimated. Often, a ``saturation''-condition is stated that should guarantee that the discretisation is ``sufficiently'' fine but it remains unclear how to check this condition in practice. We clearify, for the case of adaptive finite element methods for stationary problems, the role of the saturation assumption and how to get rigorous convergence proofs for the a posteriori refinement condition.
Multiple scales in the dynamics of
forward-backward parabolic equations
Nicola
Guglielmi (Università di L'Aquila)
We consider a non convex energy density E and direct our
attention at the formal L2-gradient system associated with the
functional $F(u) := \int_{(a,b)} E(u_x)~dx$. The parabolic equation
we obtain is ill posed because is forward or backward parabolic in
character. In order to obtain a well-posed problem we consider a
suitable regularization by adding to $F$ a convex term of higher
order, namely $\epsilon2 u_{xx}^2$, and study the regularized
gradient system for small values of $\epsilon >0$.
The scope of this talk is that of showing, with the support of
numerical simulations, various aspects of the global dynamics of
solutions and to discuss the various time (and also space) scales in
the dynamics.
This is a joint research with Giovanni Bellettini (Roma 2) and
Giorgio Fusco (L'Aquila).
On global error estimation and control for
ODEs and parabolic PDEs
Jens Lang (Technische
Universität Darmstadt)
In this talk I will report on some joint activities with Jan Verwer (Center for Mathematics and Computer Science, Amsterdam) and Kristian Debrabant (TU Darmstadt) regarding efficiency and reliability questions for initial value problems. First, systems of ODEs are considered. Existing popular codes focus on efficiency by adaptively optimizing time grids in accordance to local error control. The reliability question, that is, how large are the global errors, has received much less attention. We have implemented classical global error estimation based on the first variational equation, and global error control, for which we have used the property of tolerance proportionality. We have found, using the Runge-Kutta-Rosenbrock method ROS3P as example integrator, that the classical approach is remarkably reliable. For parabolic PDEs, the ODE approach is combined with estimates for the spatial truncation errors based on Richardson extrapolation. Numerical examples are used to illustrate the reliability of the estimation and control strategies.
Time reconstruction and a posteriori
error analysis for Galerkin and Runge-Kutta collocation
methods
Ricardo H. Nochetto (University of
Maryland, USA)
We derive a posteriori error estimates for time discretizations by both discontinuous and continuous Galerkin methods as well as by Runge-Kutta collocation methods of any order for linear and nonlinear stiff ODEs and parabolic PDEs. The key ingredients in deriving these bounds are appropriate time reconstructions of the approximate solutions and pointwise error representations. The time reconstruction is a suitable globally continuous, but locally built, piecewise polynomial function of one degree higher than the underlying time discretization, that leads to upper and lower bounds for the error regardless of the time-step; they do not hinge on asymptotics. Our estimates are optimal both globally and at the nodes, where they exhibit superconvergence order - the so-called classical order for Runge-Kutta collocation methods.
Robust a-posteriori analysis for
advection-dominated problems
Giancarlo Sangalli
(Università di Pavia)
Although the a-posteriori error analysis for elliptic problems is by now mature, the theory for advection-dominated problems is still under development. In recent years some advances have been achieved, but the developed analysis is not fully satisfactory even for the simplest one-dimensional advection-diffusion equation. This is the case I consider for the theoretical study in the present work. In spite of its simplicity, the robust a posteriori analysis of it is harder than it might seem at the first sight. In the estimate proposed, the numerical error is measured with respect to a norm which plays the role that the energy norm has with respect to the reaction-diffusion operator (i.e., the symmetric case). Numerical tests are presented in one and two space dimensions.
Adaptive, fast and oblivious convolution
quadrature
Achim Schädle (Zuse Institut
Berlin)
To approximate convolutions which occur in evolution equations with memory terms, a variable-stepsize algorithm is presented for which advancing N steps requires only O(N log N) operations and O(log N) active memory, in place of O(N^2) operations and O(N) memory for a direct implementation. A basic feature of the fast algorithm is the reduction, to differential equations which are solved numerically with adaptive step sizes. This reduction is obtained via a contour integral representation of the convolution kernel. Rather than the kernel itself, its Laplace transform is used in the algorithm. The algorithm is illustrated on an example from viscoelasticity with a fractional order constitutive law.
Adaptive space-time finite element
methods for parabolic equations
Michael Schmich
(Universität Heidelberg)
We develop an a posteriori error estimator and an adaptive algorithm for the efficient solution of parabolic partial differential equations. The error estimator assesses the discretization error with respect to a given quantity of physical interest and separates the influence of the time and space discretizations. This allows to set up an efficient adaptive strategy producing economical (locally) refined meshes for each time step and an adapted time discretization. The space and time discretization errors are equilibrated leading to an efficient method.
Adaptive FEM for a phase transition
problem with free surface flow
Alfred Schmidt
(Universität Bremen)
By heating of a wire ending, a spherical drop of molten metal
forms, which keeps its form after cooling and solidifation. This
accumulation of material can be used in a subsequent forming
process.
The melting and solidification process can be modeled by a
solid-liquid phase transition problem with free surface flow in the
melt region, varying in time. Due to the physical dimensions,
surface tension generates one of the dominant forces. Special
attention has to be payed for the space/time discretization near the
moving free boundary line of the free flow surface. We present and
study an adaptive finite element method for the coupled system.
This talk reports about joint work with Eberhard Bänsch
(Univ. Erlangen).
Convergence analysis of adaptive finite
elements for linear parabolic problems
Kunibert
G. Siebert (Universität Augsburg)
Adaptive finite elements are an efficient tool to compute
approximations to solutions of partial differential equations. They
are successfully used since the late 70th. Nowadays they are
standard tools in science and applications. Adaptive methods make
realistic simulations of multi-scale phenomena feasible, especially
in three space dimensions.
Convergence of adaptive finite element methods for elliptic problems
is now well analyzed, even optimality results are available. In
contrast to elliptic problems, there exists only the convergence
result by Chen/Jia for adaptive finite element discretizations for
parabolic problems. Based on the basic convergence result by
Morin/Siebert/Veeser for elliptic problems we will improve on the
result of Chen/Jia in several aspects: a larger problem class is
treated, the proofs are more intrinsic, and coarsening is included
in a more natural way.
This is joint work with Alfred Schmidt (ZeTeM, Universität
Bremen).
The potential for stability of
exponential Runge-Kutta methods for semi-linear
problems
Marino Zennaro (Università di
Trieste)
We consider semi-linear initial value problems for ordinary
differential equations of the type
y'(t)=Ly(t)+f(t,y(t)), t>=t_0,
with initial condition y(t_0)=y_0, where L is a constant matrix and
f(t,y) is generally nonlinear. These equations often arize from
spatial semi-discretization of parabolic problems and, typically,
the associated linear problem u'(t)=Lu(t) is stiff and, at the same
time, the nonlinear term f(t,y) is characterized by a moderate
Lipschitz constant. Moreover, the stiffness of the associated linear
problems grows as the spatial discretization becomes finer and
finer. Therefore, as far as their numerical solution is concerned,
it is important to use methods which are not too sensible to the
stiffness of the associated linear problem, i.e., which are
substantially independent of the refinement of the spatial grid. For
this purpose, a necessary feature for a method is to have good
stability properties. In this respect, a promising class of methods
is that of so-called exponential integrators. In particular, we
focus on the class of explicit exponential Runge-Kutta (EERK)
methods, which are defined by discretizing the variation-of-constant
formula. The modern techniques developed for the approximation of
the matrix exponentials made these methods competitive with
classical implicit Runge-Kutta methods. In this talk we present an
analysis of the stability properties of the EERK methods. This is
done on the basis of some suitable test equations, either in a
conditional or in an unconditional sense. The results show that the
potential of such methods in terms of the conservation of particular
contractivity properties is greater than that of classical implicit
Runge-Kutta methods.
This is a joint work with Stefano Maset from Dipartimento di
Matematica e Informatica, Università di Trieste, Italy.