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Workshop Programme

for period 7 - 11 April 2003

Multiscale Modelling, Multiresolution and Adaptivity

7 - 11 April 2003

Timetable

Monday 7 April
Session: Multiscale Modelling, Multiresolution and Adaptivity
08:30-09:55 Registration
09:55-10:00
  Opening remarks Sem 1
10:00-11:00 Brezzi, F (Pavia)
  Pseudo residual free bubbles and subgrids Sem 1
 

Residual free bubbles proved to be a powerful technique to deal with subscale phenomena. In particular they are known to provide optimal stabilizing effects in many circumstances, where traditional methods can exhibit wide oscillations or checkerboard modes.

As it is well known, however, residual free bubbles require the solution of a partial differential equation (of the same nature of the original one). This can only be done in some approximate way. These approximate solutions are often called ``pseudo residual free bubbles"

In a certain number of problems, the accuracy needed in this element by element resolution is not very high, and can be dealt with by using a very poor (but suitably chosen!) subgrid, consisting in just a few additional nodes. This occurs for instance when we want to stabilize advection-diffusion or advection-reaction-diffusion problems. In these cases we can just consider that we are adding a suitable subgrid and then solving plain Galerkin on the augmented grid (=original grid + subgrid)

In other problems there is a need to combine in an appropriate way the element subgrid with a boundary subgrid. In the boundary subgrid one can also consider non polynomial functions, such as suitable exponential or numerical solution of suitable auxiliary problems solved in small regions with a finer grid.

The lecture will present an overview of these issues and on latest developments in these directions.

 
11:00-11:30 Coffee
11:30-12:30 Weinan, E (Princeton)
  Analysis of multiscale methods Sem 1
 

We will discuss the numerical analysis of multiscale methodsfor several class of multi-physics problems

 
12:30-13:30 Lunch at Wolfson Court
14:00-15:00 Hou, T (Caltech)
  Multiscale computation for flow through heterogeneous media Sem 1
 

Many problems of fundamental and practical importance contain multiple scale solutions. Composite materials, flow and transport in porous media, and turbulent flow are examples of this type. Direct numerical simulations of these multiscale problems are extremely difficult due to the range of length scales in the underlying physical problems. Here, we introduce a dynamic multiscale method for computing nonlinear partial differential equations with multiscale solutions. The main idea is to construct semi-analytic multiscale solutions local in space and time, and use them to construct the coarse grid approximation to the global multiscale solution. Such approach overcomes the common difficulty associated with the memory effect in deriving the global averaged equations for incompressible flows with multiscale solutions. It provides an effective multiscale numerical method for computing two-phase flow and incompressible Euler and Navier-Stokes equations with multiscale solutions. In a related effort, we introduce a new class of numerical methods to solve the stochastically forced Navier-Stokes equations. We will demonstrate that our numerical method can be used to compute accurately high order statitstical quantites more efficiently than the traditional Monte-Carlo method.

 
15:00-15:30 Tea
15:30-16:30 Nochetto, R (Maryland)
  Adaptive FEM for saddle point problems: design \& convergence Sem 1
 

The lack of monotonicity in saddle point problems is a fundamental impediment to prove convergence of direct adaptive finite element methods (AFEM). We propose an alternative consisting of two nested iterations, the outer iteration being an Uzawa algorithm to update the scalar variable and the inner iteration being an elliptic AFEM for the vector variable. We show linear convergence in terms of the outer iteration counter provided the elliptic AFEM guarantees an error reduction rate together with a reduction rate of data oscillation (information missed by the underlying averaging process). We apply this idea to the Stokes system without relying on the discrete inf-sup condition; unstable elements are thus nonlinearly stabilized. We also deal with an augmented Lagrangian formulation for the Raviart-Thomas mixed finite elements, and discuss error control and error reduction rate for the corresponding elliptic operator. We finally assess complexity of the elliptic AFEM, and provide consistent computational evidence that the resulting meshes are quasi-optimal. This work is joint with E. Baensch, M. Cascon, P. Morin, and K.G. Siebert.

 
16:30-17:00 Siebert, K (Freiburg)
  Convergence of adaptive finite element methods Sem 1
 

Adaptive finite element methods (FEM) have been widely used in applications for over 20 years now. In practice, they converge starting from coarse grids, although no mathematical theory has been able to prove this assertion. Ensuring an error reduction rate based on a posteriori error estimators, together with a reduction rate of data oscillation (information missed by the underlying averaging process), we construct a simple and efficient adaptive FEM for elliptic partial differential equations. We prove that this algorithm converges with linear rate without any preliminary mesh adaptation nor explicit knowledge of constants. Any prescribed error tolerance is thus achieved in a finite number of steps. A number of numerical experiments in two and three dimensions yield quasi-optimal meshes along with a competitive performance.

 
17:00-17:45
  Informal time for researchers from various disciplines to exchange ideas \& explore currently unsolved problems Sem 1
17:45-18:30 Wine \& Beer Reception
18:45-19:30 Dinner at Wolfson Court - Residents only
Tuesday 8 April
Session: Multiscale Modelling, Multiresolution and Adaptivity
09:00-10:00 Rannacher, R (Heidelberg)
  Adaptive discretisation of optimisation problems with PDE Sem 1
 

We present a systematic approach to error control and mesh adaptivity in the numerical solution of optimization problems with PDE constraints. By the Lagrangian formalism the optimization problem is reformulated as a saddle-point boundary value problem which is discretized by a Galerkin finite element method. The accuracy of this discretization is controlled by residual-based a posteriori error estimates. The main features of this method will be illustrated by examples from optimal control of fluids and parameter estimation.

 
10:00-11:00
  Co-ordinated discussion session: Adaptivity Sem 1
11:00-11:30 Coffee
11:30-12:30 Johnson, C (Chalmers)
  Adaptive computational modelling of reaction-diffusion in laminar and turbulent incompressible flow Sem 1
 

We present a framework for adaptive computational modeling with application to reaction-diffusion in laminar and incompressible flow. We estimate using duality the total computational error in different outputs a posteriori, with contributions from both discretization in space/time and subgrid modeling of unresolved scales. We consider subgrid models based on extrapolation or local resolution of subgrid scales. We present computational results for laminar and turbulent Couette flow.

 
12:30-13:30 Lunch at Wolfson Court
14:00-15:00 Schwab, C (ETH Z\"urich)
  Finite elements for homogenisation problems Sem 1
15:00-15:30 Tea
15:30-16:00 Houston, P (Leicester)
  hp-adaptive discontinuous Galerkin methods with interior penalty for degenerate elliptic partial differential equations Sem 1
 

We consider the a posteriori and a priori error analysis of the hp-version of the discontinuous Galerkin finite element method with interior penalty for approximating second-order partial differential equations with nonnegative characteristic form. In particular, we discuss the question of error estimation for certain linear target functionals of the solution of practical interest; relevant examples include the local mean value of the field or its flux through the outflow boundary of the computational domain $\Omega$, and the evaluation of the solution at a given point in $\Omega$. Our a posteriori error bounds stem from a duality argument and include computable residual terms multiplied by local weights involving the solution of a certain dual or adjoint problem. Guided by our a posteriori error analysis, we design and implement an adaptive finite element algorithm to ensure reliable and efficient control of the error in the computed functional with respect to a user-defined tolerance. A key question in hp-adaptive algorithms is how to automatically decide when to h-refine/derefine and when to p-refine/derefine. To this end, we construct an appropriate adaptive strategy based on estimating the local Sobolev regularities of the primal and dual solutions. The performance of the resulting hp-refinement algorithm is demonstrated through a series of numerical experiments. In particular, we demonstrate the superiority of using hp-adaptive mesh refinement with the traditional h-refinement method, where the degree of the approximating polynomial is kept fixed at some low value.

 
16:00-16:30 Mackenzie, J (Strathclyde)
  Adaptive moving mesh techniques for moving boundary value problems Sem 1
 

There is considerable engineering and scientific interest in the solution of phase change problems. In this talk we consider the use of adaptive moving mesh methods to efficiently model these moving boundary value problems. These methods are based on a redistribution of mesh nodes in time while normally keeping the number and connectivity of mesh elements fixed thus leading to less complex algorithms. We consider the derivation of the moving mesh partial differential equations (MMPDEs) and appropriate adaptivity criteria. Numerical simulations of the phase-field equations and buoyancy induced heat transfer problems will be shown to demonstrate the potential of this approach which ideally should be combined with other forms of solution adaptivity.

 
16:30-17:00 Stevenson, R (Utrecht)
  An optimal adaptive finite element method Sem 1
 

We present an adaptive finite element method for solving second order elliptic equations which is (quasi-)optimal in the following sense: If the solution is such that for some s>0, the errors in energy norm of the best continuous piecewise linear approximations subordinate to any partition with N triangles are O(N^{-s}), then given an eps>0, the adaptive method produces an approximation with an error less than eps subordinate to a partition with O(eps^{-1/s}) triangles, taking only O(eps^{-1/s}) operations. Our method is based on ideas from [Binev, Dahmen and DeVore '02], who added a coarsening routine to the method from [Morin, Nochetto and Siebert '00]. Differences are that we employ non-conforming partitions, our coarsening routine is based on a transformation to a wavelet basis, all our results are valid uniformly in the size of possible jumps of the diffusion coefficients, and that we allow more general right-hand sides. All tolerances in our adaptive method depend on a posteriori estimate of the current error instead an a priori one, which can be expected to give quantitative advantages.

 
17:00-18:30
  Informal time for researchers from various disciplines to exchange ideas \& explore currently unsolved problems Sem 1
18:45-19:30 Dinner at Wolfson Court - Residents only
Wednesday 9 April
Session: Multiscale Modelling, Multiresolution and Adaptivity
09:00-10:00 DeVore, R (South Carolina)
  Tools for adaptive finite element methods Sem 1
 

We will discuss two results that have proven to be important in the analysis of convergence rates for adaptive finite element methods. The first is how to bound the number of additional subdivisions needed to remove hanging nodes in newest vertex bisection. Such a bound is necessary to control complexity bounds in adaptive methods. The second result is how to generate a near best adaptive approximation to a given function in linear time. This result has been used in coarsening routines to keep control of the number of cells in adaptively generated triangular partitions. This is joint work with Peter Binev and Wolfgang Dahmen.

 
10:00-11:00
  Co-ordinated discussion session: Wavelets and multiresolution for PDEs Sem 1
11:00-11:30 Coffee
11:30-12:30 Cohen, A (Pierre et Marie Curie)
  Multiscale adaptive schemes for hyperbolic equations: some convergence results Sem 1
 

we shall discuss the convergence analysis of a class of adaptive schemes for evolution equations. These schemes combine adaptive mesh refinements with wavelets based on a framework introduced by Ami Harten. The analysis will mainly focus on the hyperbolic case.

 
12:30-13:30 Lunch at Wolfson Court
14:00-15:00 Dahmen, W (RWTH Aachen)
  Adaptive multiscale application of operators Sem 1
 

This talk is concerned with the adaptive application of (linear and nonlinear) operators in wavelet coordinates. There are two principal steps, namely first the reliable prediction of significant wavelet coefficients of the result of such applications from those of the input, and second the accurate and efficient computation of the significant output coefficients. Some of the main conceptual ingredients of such schemes are discussed. Moreover, we indicate how this leads to adaptive solution schemes for variational problems that can be shown to have asymptotically optimal complexity. This is joint work with A. Cohen and R. DeVore.

 
15:00-15:30 Tea
15:30-16:00 Kunoth, A (Bonn)
  Adaptive wavelet methods for control problems with PDE constraints Sem 1
 

For the fast numerical solution of a control problem governed by a stationary PDE with distributed or Neumann boundary control, an adaptive algorithm is proposed based on wavelets. A quadratic cost functional involving natural norms of the state and the control is to be minimized subject to constraints in weak form. Placing the problem into the framework of (biorthogonal) wavelets allows to formulate the functional and the constraints equivalently in terms of sequence norms of wavelet expansion coefficients and constraints in form of an automorphism. The resulting first order necessary conditions are then derived as a (still infinite) weakly coupled system of equations. Once this system is obtained, the machinery developed by Cohen, Dahmen and DeVore can be employed for the design of an adaptive method which can be interpreted as an inexact gradient method. In each iteration step the primal and the dual system needs to be solved up to a prescribed accuracy. In particular, we show that the adaptive algorithm is asymptotically optimal, that is, the convergence rate achieved for computing the solution up to the target tolerance is asymptotically the same as given by the wavelet-best N-term approximation of the solution, and the total computational work is proportional to the number of unknowns.

 
16:00-16:30 Urban, K (Ulm)
  Adaptive convex optimisation: convergence results Sem 1
 

We consider the minimization of a convex functional on reflexive Banach spaces. We introduce an infinite-dimensional (exact) algorithm of steepest descent type and show its convergence. Next we consider minimization problems that can be transformed into \ell_p-spaces. In this case, we formulate a computable version of the adaptive algortihm and show its convergence. We present joint work with Claudio Canuto.

 
16:30-17:00 Hawkins, SC (Liverpool)
  Easily inverted approximation type preconditioners for almost-diagonal matrices arising from two-dimensional elliptic operators Sem 1
 

For elliptic operators represented in a wavelet basis there exists an optimal diagonal preconditioner that yields a condition number that is independent of basis size. This is well known.But in practice the condition number can still be large and this must be addressed by further preconditioning. Preconditioners based on Sparse Approximate Inverse techniques have been presented by Cohen and Masson [Siam J. Sci. Comput., 21:1006--1026, 1999], and by Chan, Tang and Wan [BIT, 37:644--660, 1997] for matrices obtained by applying a discrete wavelettransform to a matrix. Preconditioning with an easily inverted sparse approximation is considered in the one-dimensional case by Chen [ETNA, 8:138--153, 1999]. We consider easily inverted sparse approximation preconditioning in the two-dimensional case. While representation in a single scale basis typically produces a matrix with diagonal structure, representation in a wavelet basis produces a more complicated structure sometimes called almost-diagonal, or finger-patterned in the case of one-dimensional problems. Sparse matrices with these structures typically have dense LU factors, and so are difficult to invert. Chen reorders the wavelet basis for one-dimensional problems to produce what is essentially a banded matrix---the banded part of the matrix is the desired easily inverted sparse approximation because it has sparse factors that can be cheaply computed. The ordering concentrates much of the finger-patterned structure into the banded matrix so that it makes a good preconditioner. Preconditioners of this type have not previously been developed for two-dimensional problems.

 
17:00-18:30
  Informal time for researchers from various disciplines to exchange ideas \& explore currently unsolved problems Sem 1
18:45-19:30 Dinner at Wolfson Court - Residents only
Thursday 10 April
Session: Multiscale Modelling, Multiresolution and Adaptivity
09:00-10:00 Canuto, C (Torino)
  Wavelet-based adaptive optimisation Sem 1
 

We use the properties of wavelets bases to design an adaptive descent algorithm for solving a convex minimization problem in a function space of Hilbert type. We prove the convergence of the algorithm and we discuss its optimality in the framework of best N-term approximation results (Cohen, Dahmen and DeVore). This is a joint work with K. Urban.

 
10:00-11:00
  Co-ordinated discussion session: Multiscale methods for PDEs Sem 1
11:00-11:30 Coffee
11:30-12:30 Ainsworth, M (Strathclyde)
  hp-finite adaptive element methods for time harmonic Maxwell's equations Sem 1
 

Recently, there has been considerable interest in the use of high order finite element methods for the approximation of Maxwell's equations. We shall survey some of our own work in this area. In particular, we shall present families of hierarchic basis functions for the Galerkin discretisation of the space $H({\rm curl};\Omega)$ that naturally arises in the variational formulation of Maxwell equations. The conditioning and dispersive behaviour of the elements is discussed along with approximation theory. Numerical examples are shown which indicate the potential of the methods for computing approximations of the time-harmonic Maxwell's equations.

 
12:30-13:30 Lunch at Wolfson Court
14:00-15:00 Carstensen, C (Vienna)
  Scientific computing of related minimisation models in three applications Sem 1
 

The three applications considered for a nonconvex minimisation problem model examples for phase transitions, optimal design tasks, and micromagnetics. Since (quasi-) convexity is essentially equivalent to lower semicontinuity, typically, the minimiser in such nonconvex minimisation problems is NOT attained: Infimising sequences develop enforced oscillations of an arbitrary fine length scale and converge weakly but not strongly to an averaged generalised solution. The approximation of which plus an approximation of the generated Young measures is the task of a numerical simulation. In the direct approach to computational microstructures, the finite element solutions develop oscillations quite difficult to compute. In the three examples at hand, relaxation theory provides auxiliary problems (relaxed models) which are convex but not uniformly or even strictly convex. Their numerical simulation, in the quasiconvexified approach to computational microstructures, is relatively easy and recommended and even the Young Measures generated in the nonconvex model can be recovered. However, there is a reliability-efficiency gap and strong convergence in energy norms is under debate when stabilisation techniques are employed.

 
15:00-15:30 Tea
15:30-16:00 Farmer, C (Schlumberger)
  Geological modelling: multiphysics data to multiscale models Sem 1
 

The talk discusses competing approaches to modelling multiscale systems regarding flow through porous media.It will be shown that 'upscaling' is in somecircumstances a multiscale finite volume method.

 
16:00-16:30 Moon, K-S (RIT)
  Adaptive Monte Carlo algorithm for killed diffusion Sem 1
16:30-17:00 von Schwerin, E (KTH)
  Convergence rates for adaptive FEM Sem 1
17:00-18:30
  Informal time for researchers from various disciplines to exchange ideas \& explore currently unsolved problems Sem 1
20:00-00:00 Conference Dinner - Selwyn College (Dining Hall)
Friday 11 April
Session: Multiscale Modelling, Multiresolution and Adaptivity
09:00-10:00 Xu, J (Penn State)
  Multilevel techniques for grid adaptation Sem 1
 

Some recent studies will be reported in this talk on using multigrid ideas in grid adaptation. Results to be presented include gradient and Hessian recovery schemes by using averaging and smoothing (as in multigrid), interpolation error estimates for both isotropic and anisotropic grids and multilevel techniques for global grid moving and local grid refining. Some applications will also be given. (Parts of the talk are joint works with Randy Bank, and with Long Chen and Pengtao Sun).

 
10:00-11:00
  Co-ordinated discussion session: Multigrid and multilevel methods Sem 1
11:00-11:30 Coffee
11:30-12:00 Barth, TJ (NASA)
  Preliminary results for a-posteriori error estimation and adaptivity in numerical magnetohydrodynamics Sem 1
12:00-12:30 Jimack, P (Leeds)
  Anisotropic mesh refinement based upon error gradients Sem 1
 

Many multiscale phenomena are highly anisotropic. For example planar shocks, boundary layers, solidification fronts, etc., each have particular directions for which the resolution should be finest (i.e. perpendicular to the shock/layer/front). In order to obtain an adaptive algorithm of optimal complexity therefore the refinement must respect this local directionality. This requires two separate, but equally important, components: a practical mesh adaptivity mechanism for delivering a desired local anisotropic mesh refinement; and an error estimator that can efficiently and reliably direct this refinement. This presentation will focus on the second of these two components by considering a strategy for driving anisotropic mesh refinement that is based upon the use the gradient of the error function, or a suitable approximation to it. Unlike with conventional error estimation, which is only designed to indicate where it is necessary to locally refine, for an estimate to be useful for anisotropic refinement it must also provide information on which direction(s) in which to refine. It will be demonstrated that the use of anisotropic adaptivity driven by the gradients of certain a posteriori error estimates is indeed feasible and appears to offer significant potential in terms of efficiency gains that move towards an optimal-complexity algorithm, at least for a class of linear reaction-diffusion test problems that is considered.

 
12:30-13:30 Lunch at Wolfson Court
14:00-15:00 Engquist, B (Princeton)
  Heterogeneous multiscale methods for dynamical systems Sem 1
15:00-15:30
  Closing discussion Sem 1
18:45-19:30 Dinner at Wolfson Court - Residents only

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