AARMS (the Atlantic Association for Research in the Mathematical Sciences), the AARMS Collaborative Research Group in Numerical Analysis and Scientific Computing, the CRM (Centre de Recherches Mathématiques), and the Department of Mathematics and Statistics at Memorial University will bring researchers interested in adaptive methods for partial differential equations to a 5-day workshop in beautiful St. John's, NL, Canada.
The meeting will consist of a 2-day short course on adaptive methods for PDEs taught by leading expert Dr. Weizhang Huang (University of Kansas), research level talks by experts in the area of adaptivity and by academic and industrial researchers with applied problems who have an interest in exploring adaptive numerical techniques. The final part of the workshop will be interactive, linking the applied researchers with adaptivity experts to investigate the process of introducing adaptivity to their simulations.
A posteriori error estimation for conforming, non-conforming, mixed and discontinuous finite element schemes are discussed within a single framework. By dealing with four ostensibly different schemes under the same umbrella, the same common underlying principles at work in each case are highlighted leading to a clearer understanding of the issues involved. The ideas are presented in the context of piecewise affine finite element approximation of a second-order elliptic problem. In all cases one has computable upper bounds on the error measured in the energy norm and corresponding local lower bounds showing the efficiency of the schemes. We present numerical examples illustrating how the estimators may be used to control adaptive mesh refinements for finite element approximation of a some representative partial differential equations efficiently.
Optimal transport methods redistribute a mesh using a combination of a scalar monitor function to control the mesh density and a global regularity constraint. These methods are easy and cheap to implement. I will show in this talk that they are also provably effective in generating anisotropic meshes for problems with both linear features and features with high curvature. This leads to good error estimates when approximating functions. I will illustrate this by considering the use of these methods for meteorological examples both in numerical weather prediction and also in data assimilation.
Anisotropic eigenvalue problems arise in the application of the Laplace-Beltrami operator to geometric shape analysis and imaging segmentation. They also arise from stability or sensitivity analysis and construction of special solutions such as traveling wave and standing wave solutions for anisotropic diffusion partial differential equations, which occur in many areas of science and engineering including plasma physics, petroleum engineering, and image processing. This talk is concerned with finite element computation of those eigenvalue problems. Emphases will be on anisotropic mesh adaptation and preservation of basic structures. Both analytical analysis and numerical results will be presented.
Hessian recovery has been widely used in anisotropic mesh adaptation for directional information of the solution error. Unfortunately, for linear finite elements, a recovered Hessian is non-convergent in general, although it has been observed numerically that adaptive meshes based on such a non-convergent approximation nevertheless lead to an optimal error. We discuss an error bound for the linear finite element solution of a general boundary value problem under a mild assumption on the closeness of the recovered Hessian to the exact one. Numerical results show that this closeness assumption is satisfied by the approximated Hessian obtained with commonly used Hessian recovery methods. Moreover, we will see that the bound on the finite element error changes gradually with the closeness of the recovered Hessian.
Electromagnetic (EM) methods have been used extensively for detecting and delineating natural resources including metallic ore deposits, groundwater, and hydrocarbon reservoirs. Quantitative interpretation of EM survey data -- how big, how valuable, how deep is the resource? -- involves constructing a model of the Earth such that EM data synthesized for this prospective Earth model adequately match the measured data. The ability to compute the electric and magnetic fields in the Earth model due to the particular source used in a survey is a critical component of this interpretation process. This talk will give an introduction to geophysical EM methods, and then attempt to describe and illustrate the challenges involved in synthesizing geophysical EM survey data.
Reaction diffusion equations naturally give rise to solutions with highly localized structures. Such systems are typically used to model the formation of isolated structures in developmental biology. Such structures result in different spatial scales in the solution. The dynamics of such isolated structures typically evolves on a slow timescale. In some cases an exponentially slow time scale. However, a bifurcation or interaction with the domain may result in a faster temporal evolution superimposed on the systems slow dynamics. In addition, we are now considering the addition to delay terms to all such systems. Although we can often obtain approximate solutions using asymptotic analysis, it is difficult to verify these results with a comparison to full numerical results.
When modeling the flows of classical or complex fluids, a common challenge is the accurate modeling of boundary and interior layers that determine many of the critical properties of the flow but occupy a vanishingly small volume. To gain better understanding of these phenomena, we can also consider classical singularly perturbed problems of reaction-diffusion type, which exhibit the same boundary and interior layers within a much simpler mathematical (and numerical) setting. In cases where the locations of these layers are known, robust numerical techniques exist for a priori adaptation of meshes to resolve the layers with accuracy independent of the singular perturbation parameters. We have recently developed a Petrov-Galerkin based finite-element method for singularly perturbed reaction diffusion equations that offers coercivity and continuity in a so-called balanced norm, which offers much better approximation than traditional Galerkin techniques. In this talk, we will give an introduction to this approach and highlight the challenges in coupling it with a parameter-robust adaptive meshing technique.
In current health care, digital imaging plays a vital role in many clinical settings from disease diagnosis and monitoring to lesion quantification, treatment assessment and surgical planning. Manually processing a large amount of images can be very time-consuming for radiologists. Hence computer-aided diagnostic (CAD) medical systems have become necessary tools to assist radiologists in their medical decision-making processes. CAD technologies have been used in radiology for more than two decades and their efficacy has been proven promising. However, presence of noise, limitation of imaging hardware, subtle and irregular shaped lesions are still big hassles. In this talk, we will discuss two important issues, i.e., de-noising and segmentation, and their challenges in CAD applications.
Kanschat and Riviere has recently proposed an H(div) conforming mixed finite element method for the coupled Darcy-Stokes flow, which uses a unified discretization for both the Stokes side and the Darcy side. In their method, the discrete velocity field is continuous in the normal direction across the Stokes-Darcy interface, which we refer to as a "strong coupling". Many other numerical methods use a "weak coupling" scheme. That is, the normal continuity of the velocity field across the Darcy-Stokes interface is imposed weakly by introducing a Lagrange multiplier. Here, we develop an a posteriori error estimator for the H(div) conforming, "strongly coupled" mixed formulation. Due to the strong coupling on the interface, special techniques need to be employed in the proof of its global reliability and efficiency. This is the main difference between our work and the previous work by Babuska and Gatica (2010), in which they considered a weakly coupled interface condition.
In this paper, we develop an efficient moving mesh method for equilibrium radiation diffusion equations. The method is based on the quasi-Lagrange approach of moving mesh strategy with which the mesh is considered to move continuously in time. Several issues arising from the implementation of the scheme, including variational mesh generation, ``freezing coefficient'' method to linearize the nonlinear diffusion coefficient and ``cut-off'' method to ensure the nonnegativity of energy density is addressed. Particularly, it is found that a two-level moving mesh method can save much more CPU time than one-level moving mesh method while still keeping the same accuracy. Numerical results show that our moving mesh method can be used to multi-material, multiple spots concentration situation of two-dimensional radiation diffusion models.
Anisotropic diffusion problems arise in many fields of science and engineering. Non-physical solution may appear in the numerical computations when standard discretization methods and regular meshes are used to solve the problems. In those cases, the numerical solution is said to violate the discrete maximum principle (DMP). There are two general approaches to improve the numerical solutions to satisfy DMP. One is to develop proper discretization schemes and the other is to utilize proper meshes. In this talk, I will present mesh adaptation method based on anisotropic metric tensors that are used to generate M-uniform meshes. We developed a particular metric tensor such that the linear finite element approximations based on the corresponding mesh are guaranteed to satisfy DMP. In addition, we also developed metric tensor based on DMP satisfaction together with error estimate. The corresponding mesh not only guarantees the satisfaction of DMP but also reduces the interpolation error. Some numerical results for 2D and 3D anisotropic diffusion problems will be presented.
There are three ways to participate in the meeting
Registration is now live at this link . Registration fees are set at $150 for faculty or participants from industry, $100 for post-doctoral researchers, $65 for undergraduate or graduate students. The registration fee includes the adaptivity short course, access to all talks, all breaks and both receptions (see social events listed below!). NOTE: a $25 cancellation fee applies to all cancellations/refunds. Please contact me if you have questions about the workshop before you register.
List of Registered Participants
A opening reception will be held outside of Room 1043 in the Arts and Admin Building on MUN's St. John's campus on Sunday August 17 @ 7:00pm. This reception is included with your registration. On Thursday August 21 we will have a social event starting at 5pm at O'Reilly's pub on George Street in downtown St. John's. The cost of this reception is included with your paid registration. You are responsible to find your own way to and from, here is a walking map or you can cab-pool, the fare should be less than $15. Additional tickets can be purchased at the time of registration.
The university dorms are the most economical choice for accommodations, see the first link below. There is only one dorm option: the new residence rooms, please download, print, fill-out and email the following form to the email address on the form (Not to me!!)
We have also reserved a small block of rooms at the Guvnor Inn (second link below), the booking code MUNMath2014 is now active. This will get you access to rooms for $119 per night. Please call the hotel directly to take advantage of this rate, it will not work on the website I am told. This block of rooms will be released about 30 days before the workshop.
Everyone is responsible for booking and paying for their own accomodations! Here are a list of some possibilities. The first two are within easy walking distance to campus and the workshop, the third about a 20 min walk, the other two further away. The Quality Inn is downtown if you would rather stay there. Expedia, hotels.com will give other options of course!
Book your flight into St. John's International Airport YYT: St. John's, NL (not to be confused with YSJ: St John, NB!)
Note: there is no shuttle from YYT to the hotels. Taxis are readily available with fixed fares ($25-30) varying according to distance between the airport and your destination. Taxis are available just outside the obvious exit as you leave the baggage area.
Here's a map.
If you are staying on campus you will be provided with a wireless internet password. If your home campus participates in Eduroam you likely will be able to connect at Memorial U - see details here. If neither of the previous two apply (or if the latter applies and you don't want to take any chances) please email me to indicate your interest and I will set you up with a wireless password.
There's lots to see in and around St. John's:
Atlantic Association for Research in the Mathematical Sciences
Centre de Recherches Mathematiques
National Science Foundation (US)
Dept of Mathematics and Statistics, MUN
Office of the Vice President - Research, MUN