CIMPA - NPDE - NBHM Research School on
"Current Trends in Computational Methods for PDEs"
Dates: July 8-19, 2013
Venue: Department of Mathematics , Indian Institute of Science Bangalore , India
Last Date for Application : March 24th, 2013
This CIMPA research school is an advanced level workshop focusing on the current topics in computational methods for partial differential equations. The goal is to train final year M. Sc students, students pursuing PhD, post doctoral researchers and young faculty in the research area. The Research school will consist of participants from India (25 participants), from neibouring countries to India (15 particiapnts) and from developed countries (10 participants). (This webpage will be updated continuously).
Funded by CIMPA, CLAY MATH INSTITUTE, DST (NPDE-TCA), ICERM-VIMSS, ICTP, IMI-IISC, IMU, NBHM, UNESCO
Lectures Venue: Lecture Hall-I, Department of Mathematics, IISc Bangalore
CIMPA Research School Schedule (July 8-19, 2013)

## CIMPA School Photos

 Coordinators : Blanca Ayuso de Dios, Centre de Recerca Matematica (CRM), Barcelona, Spain Thirupathi Gudi, Indian Institute of Science Bangalore, India Local Organizing Committee : Blanca Ayuso de Dios, CRM, Barcelona, Spain Sashikumar Ganesan, IISc Bangalore, India Thirupathi Gudi, IISc Banglaore, India A. K. Nandakumaran, IISc Bangalore, India Neela Nataraj, IIT Bombay, India Courses (All the Courses will be taught in English): Adaptive Finite element methods C^0 Interior Penalty Methods Finite elements for mixed variational formulations A course related to applications Numerical methods for non-linear hyperbolic PDEs Numerical techniques for PDEs with random input data Speakers : Bobby Philip, Oak Ridge National Laboratory Daniele Boffi , Universita' degli Studi di Pavia, Italy Susanne C. Brenner , Louisiana State University, USA Lucia Gastaldi , Universita' di Brescia, Italy Ulrik Skre Fjordholm , Eidgenössische Technische Hochschule(ETH), Switzerland Fabio Nobile , Ecole Polytechnique Federale de Lausanne (EPFL), Switzerland Li-yeng Sung , Louisiana State University, USA Raul Tempone , King Abdullah University of Science and Technology (KAUST), Saudi Arabia Andreas Veeser , Università degli Studi di Milano, Italy Information for Participants: Financial Support: Participants from India will be provided free Boarding, Lodging and at most a three tier A/c train fare by the shortest route. Participants from the developing nations will be provided free Boarding, Lodging and up to Eur 400 (or the actual fare if it is less than EUR 400) travel support towards airfare. Participants from developed nations will pay on their own for their Boarding, Lodging and Travel. Accommodation: Participants will be accommodated in the Centenary Visitors House (CVH), IISc Banglore. IISc Campus Map , (See 4 in 3B in the map) Arrival and Departure : Arrival is on or later than June 23, 2013. Departure should be on or before July 20, 2013.

 Description of the Courses: Adaptive Finite element methods (Speaker: Andreas Veeser): adaptive finite element methods are a widely used tool to approximate solutions of partial differential equations and, thus, to solve computationally many problems in science and engineering. Adaptivity improves the exploitation of the computational resources and is often crucial for practical solvability, especially in 3d problems. It consists of an iterative process that tunes, e.g., the underlying mesh by extracting information on the error from the current approximate solution. The latter is done with the help of so-called a posteriori error estimators. A relatively mature theory for the design of a posteriori estimators has been developed in the 1980s and 1990s, while most theoretical results about adaptivity itself are more recent. The following course is partially based on previous courses at summer schools in Germany and Italy (CIME) and emphasizes theoretical aspects, including adaptive approximation. The course starts with a model adaptive finite element method for solving an elliptic boundary value problem. It then gives a brief account on the theory of a posteriori error estimators, emphasing the concepts 'residual' and `oscillation'. The analysis of the adaptive algorithm starts by proving that the sequence of generated finite element solutions converges to the exact solution of the boundary values problem. The remaining part of the course provides a quantitative variant of this result. First, convergence rates are established in the simpler case when the target function is explicitly known. This prepares the general case when the target function is an unknown solution of a boundary value problem and in particular reveals the origins of the assumptions to be required. The proper derivation of the convergence rates for the adaptive finite element method includes a discussion of the contraction property and the cost of keeping conformity under refinement. C^0 Interior Penalty Methods (Speakers: Susanne C. Brenner and Li-yeng Sung): C^0 interior penalty methods are discontinuous Galerkin methods for fourth order elliptic boundary value problems that use standard Lagrange finite element spaces for second order problems. They are simpler than $C^1$ finite element methods and, unlike classical nonconforming finite element methods, they come in a natural hierarchy so that smooth solutions can be captured efficiently by higher order methods. Moreover, unlike mixed finite element methods, $C^0$ interior penalty methods preserve the symmetric positive definiteness of the continuous problems. In this course we will (i) derive these methods, (ii) present {\em a priori} and {\em a posteriori} error estimates, and (iii) construct and analyze multigrid and domain decomposition algorithms for the resulting discrete problems. Finite elements for mixed variational formulations (Speakers: Daniele Boffi and Lucia Gastaldi): In the first part of the course we discuss the finite element approximation of mixed formulations. It is well known that the finite element spaces should satisfy suitable inf-sup conditions for the stability and the convergence of the discrete scheme. The theory is applied to second order elliptic problems and Stokes equations, for which an overview of standard discretizations is presented. The second part of the course is devoted to the study of the approximation of eigenvalue problems. In the case of Galerkin discretizations of elliptic partial differential equations, the conditions for the convergence of eigenvalues and eigenfunctions are the same as for the convergence of the solutions to the corresponding source problem. We review the basic arguments of the analysis which is often referred to as Babuska-Osborn theory. In the case of mixed formulations, however, it will be shown that standard conditions for the source problem (like the well-known Babuska-Brezzi inf-sup condition) are not sufficient for the good convergence of eigenvalues and eigenfunctions. Finally, we shall discuss the numerical approximation of eigenvalue problems in the setting of differential forms. This includes the study of the Hodge-Laplace eigenvalue problem in the framework of de Rham complex. The abstract theory is applied to the approximation of Maxwell's equations. A Course Related Applications (Speaker: Bobby Philip): Numerical methods for non-linear hyperbolic PDEs (Speaker: Ulrik Skre Fjordholm): Many models in physics and engineering involve non-linear hyperbolic equations. Examples include the shallow-water equations of oceanography, Euler equations of aerodynamics, Magneto-Hydrodynamic(MHD) equations of plasma physics and equations of non-linear elasticity. The solutions of these equations contain discontinuities in the form of shock waves and other interesting flow features.\\ The envisaged lecture course will present both basic as well as state of the art numerical methods to simulate these equations. We will present a brief overview of the theory for the continuous problem. We start with standard monotone schemes for scalar conservation laws and describe approximate Riemann solvers as well as central schemes for systems of conservation laws. High-order finite volume schemes based on non-oscillatory reconstruction procedures will be presented.We conclude with a brief description of very high-order methods like ENO, WENO and Discontinuous Galerkin (DG) methods. If time permits, some realistic applications to oceanography and astrophysics will also be presented. Numerical techniques for PDEs with random input data (Speakers: Fabio Nobile and Raul Tempone) When building a mathematical model to describe the behavior of a physical system, one has often to face a certain level of uncertainty in the proper characterization of the model parameters and input data. Examples appear in the description of flows in porous media, behavior of living tissues, combustion problems, deformation of composite materials, meteorology and atmospheric models, etc. The increasing computer power and the need for reliable predictions have pushed researchers to include uncertainty models, often in a probabilistic setting, for the input parameters of otherwise deterministic mathematical models. In this series of lectures we focus on mathematical models mostly based on Partial Differential Equations with stochastic input parameters (coefficients, forcing terms, boundary conditions, shape of the physical domain, etc.), and review the most used numerical techniques for propagating the input random data onto the solution of the problem.
Pre-School to the Research School
An instructional pre-school will precede the CIMPA Research School which will be focused on prerequisites and basic training for the participants of the CIMPA Research School.
Dates for Pre-School : June 24-July 07, 2013 .
Venue : Department of Mathematics, Indian Institute of Science Bangalore .
All the participants in the CIMPA School are recommended to participate in the Pre-School.

## Preschool Material

Speakers and Courses (Tentative) for the Pre-School: (Courses in the Pre-school are Introductory)
Blanca Ayuso de Dios (CRM, Barcelona) : Basics in Probability Theory and Multilevel Methods
Sashikumaar Ganesan (IISc Bangalore) : Finite Element Methods for Stokes Problem
G. D. Veerappa Gowda (TIFR-CAM Bangalore) : Numerical Methods for Hyperbolic Problems
Thirupathi Gudi (IISc Banglaore) : Adaptive FEM
A. K. Nandakumaran (IISc Bangalore) : Weak Formulations/Advanced PDEs
Neela Nataraj (IIT Bombay) : Introduction to Finite Element Methods
Amiya K. Pani (IIT Bombay) : Discontinuous Galerkin FEM
Phoolan Prasad (IISc Bangalore) : Theory of First Order PDEs
Mythily Ramaswamy (TIFR-CAM Bangalore) : Theory of Distributions/Sobolev Spaces