IST - Partial Differential Equations : From Theory to Computation (2019)

The list of speakers and course associates is given below

 Syllabus:

 Dr. T. Muthukumar from IIT Kanpur delivered six lectures of 90 minutes each on Variational Approach for PDEs. First, he gave a quick review of distributions, mainly the concept of weak derivative, Fourier transform and convolution. After that he discuss the Sobolev spaces; mainly H^1 space and important inequalities, imbedding and Trace theorems. Then, variational approach to elliptic PDEs, was discussed along with existence, uniqueness, regularity results and eigenvalue problems. The following topics were discussed during the lectures and corresponding problems were solved during tutorial classes;

1 Theory of Distribution

• 1.1 Need to relax the notion of Solution of a DE
• 1.2 Space of Test Functions
• 1.2.1 Smooth Non-Analytic (Bump) Functions
• 1.2.2 Mollifiers
• 1.2.3 Inductive Limit Topology on Cc  ()
• 1.2.4 Regularization and Cut-off Technique
• 1.3 Space of Distributions
  • 1.3.1 Functions as Distributions
  • 1.3.2 Measures as Distributions
  • 1.3.3 Multipole Distributions
  • 1.3.4 Infinite Order Distributions
  • 1.3.5 Topology on Distributions

• 1.4 Operations with Distributions
• 1.4.1 Differentiation

2 Sobolev Spaces

• 2.1 Wk;p(Ω) Spaces
• 2.2 Smooth Approximations
• 2.3 Characterisation of W k , p ()
• 2.3.1 Poincar e Inequality

• 2.4 Extension Operators
• 2.5 Topological Dual of Sobolev Spaces
• 2.6 Fractional Order Sobolev Space
• 2.7 Imbedding Results
  • 2.7.1 Sobolev Inequality (1 ≤ p < n)
  • 2.7.2 Equality case, p = n
  • 2.7.3 Morrey’s Inequality (n < p <  )
  • 2.7.4 Generalised Sobolev Imbedding
  • 2.7.5 Compact Imbedding
• 2.8 Trace Theory
• 3 Variational Inequality
• 4 Second Order Elliptic PDE
  • 4.1 Homogeneous Dirichlet Problem
  • 4.2 Inhomogeneous Dirichlet problem
  • 4.3 Linear Elliptic Operators
  • 4.4 Periodic Boundary Conditions
  • 4.5 Coercive Neumann Problem
  • 4.6 Semi-coercive Neumann Problem
  • 4.7 Mixed and Robin Problem
  Appendices

Prof. K. Sreenadh from IIT Delhi delivered six lectures of 90 minutes each on Classical PDEs. He basically discussed the Laplace equation, Harmonic functions, maximum principle, fundamental solution, Green’s function and the solvability of Dirichlet problem. After that Prof. Sreenadh discussed the heat equation in detail with weak and strong maximum principles for the heat equation, fundamental solution, existence and uniqueness for the initial value problem. The following topics were discussed during the lectures and corresponding problems were solved during tutorial classes;

1 Elliptic Equations

• 1.1 Potential Theory
• 1.1.1 Fundamental Solution
• 1.1.2 Dirichlet Problem
• 1.1.3 Properties of Harmonic Function
• 1.1.4 Regularization and Cut-off Technique
• 1.2 Maximum Principles
• 1.3 Existence of Harmonic Functions: Perron’s Method
• 1.4 Eigenvalue Problems
• 1.5 Problems

Dr. Rajendra K. Ray from IIT Mandi delivered six lectures of 90 minutes each on Finite Difference Methods. He first discussed the basics of numerical computations and different examples in the field of fluid dynamics, solid mechanics, and other related areas to grow the interest among the participants in this topic. After that the basic approaches of finite difference approximations of first and second order derivatives were discussed. Then those approximations were used to discretize some differential equations. After that the consistency, convergence and stability for general finite difference methods were discussed. Finally, he discussed the stability and error
estimates of different finite difference methods and its usefulness to discretize the following equation:

• Heat equation (1D and 2D)
• Wave equation (1D and 2D)
• Laplace equation (1D and 2D)

After discussing the above three equations representing three difference classes, some nonlinear equations like semi-linear heat equation, Burger’s equation and unsteady convection-diffusion equation were also discussed. At the end some advance numerical methods like higher order accurate (HOC) finite difference scheme and advance iterative solver like SOR, Bi-Conjugate gradient algorithm was discussed and highlighted some new phenomenon discovered through numerics from the speaker’s own research work. The following topics were discussed during the lectures and corresponding problems were solved during tutorial/computer lab classes;

1 Introduction

• 1.1 Characteristics of Numerical Computing
• 1.2 Partial Differential Equations and their Classifications
• 1.2.1 Physical Classification
• 1.2.2 Mathematical Classification

2 Finite Difference approximations for Partial Derivatives

• 2.1 Simple Method
• 2.2 General Method
• 2.3 Polynomial Fitting
• 2.4 Non-uniform Mesh

3 Accuracy of Finite Difference Solution

• 3.1 Consistency
• 3.2 Stability
• 3.3 Convergence

4 Finite Difference Schemes

• 4.1 Heat equation
• 4.2 Wave equation
• 4.3 Laplace equation

5 Some Nonlinear Equations

5.1 Convection Diffusion Equation
5.2 Burgers Equation

Dr. Saumya Bajpai from IIT Goa delivered six lectures of 90 minutes each on Finite Element Methods. First she introduce the basic idea behind the development of finite element methods and the construction of finite element spaces. After that she discussed the interpolation theory and the applications of finite element method to discretize and solve elliptic and parabolic PDEs. Finally, she discussed the error
analysis and some implementation aspects. The following topics were discussed during the lectures and corresponding problems were solved during tutorial/computer lab classes;

1 Introduction

• 1.1 Background
• 1.2 Finite Difference Methods (FDM)
• 1.3 Finite Element Methods (FEM)

2 Theory of Distribution and Sobolev Spaces

• 2.1 Review of L2- Space
• 2.2 A Quick Tour to the Theory of Distributions
• 2.3 Elements of Sobolev Spaces

3 Abstract Elliptic Theory
3.1 Abstract Variational Formulation
3.2 Some Examples

4 Elliptic Equations
4.1 Finite Dimensional Approximation to the Abstract Variational Problems
4.2 Examples
4.3 Computational Issues
4.4 Adaptive Methods

Bibliography

Dr. Manotosh Kumbhakar from IIT Mandi assisted in fifteen tutorials of one hour each. Besides this he also attended all the lectures and helped participants with their doubts after the classes and tutorials. The participants
connected with Dr. Manotosh quite naturally and his presence was of a great help to the participants. They learnt a lot from him. He added a lot of value to the IST.

Mr. Sudeb Majee from IIT Mandi assisted in fifteen tutorials of one hour each. Besides this he also attended all the lectures and helped participants with their doubts after the classes and tutorials. The participants connected with Mr. Sudeb quite naturally and his presence was of a great help to the participants. They learnt a lot from him. He added a lot of value to the IST.

 Time Table
Tea and Snack Breaks: 11.00 - 11.30 and 4.00 - 4.30
Lunch Break: 1.00 - 2.00

TM: T. Muthukumar
KS: K. Sreenadh
RR: Rajendra K. Ray
SB: Saumya Bajpai
SM: Sudeb Majee
MK: Manotosh Kumbhakar

List of actual Participants

How to reach

Click here