1. Partial Derivatives Example: Find If solution: Partial Derivatives Example: Find If solution: gradient grad(u) = gradient

2. Partial Derivatives Example: Find divergence of If solution: divergence Example: Find If solution: laplacian Laplacian of u = div(v)

3. Partial Differential Equations (PDE) Definition: a partial differential equation (PDE) is an equation that contains partial derivatives 1 4 7 2 5 6 3

4. Partial Differential Equations (PDE) Order of a PDE: 1 4 7 6 3 The order of a differential equation (ODE or PDE) is the order of the highest derivative in the equation.

5. Linear 2ed-Order PDE The general linear 2ed order PDE in two variables x, y. The discriminant of the equation = Definition:

6. Linear 2ed-Order PDE (Classification) The general linear 2ed order PDE in two variables x, y.

7. Linear 2ed-Order PDE

8. Definitions and Terminology Definition: Solution of PDE Any function which when substituted into a PDE reduces the equation to an identity, is said to be a solution of the equation. Solution of a Partial Differential Equation Can you think of another solution ??? A solution of a PDE is generally not unique

9. Definitions and Terminology Boundary Condition This PDE has an infinite number of solutions D BVP: Boundary Value Problem

10. Definitions and Terminology D BVP: Boundary Value Problem D Find a function which satisfy the PDE inside the domain and it assumes given values on the boundary BVP: Boundary Value Problem Dirichlet Boundary Condition

11. Definitions and Terminology D Find a function which satisfy the PDE inside the domain and it assumes given values on the boundary BVP: Boundary Value Problem the boundary consists of two Vertical lines and two horizontal lines where Dirichlet Boundary Condition

12. Definitions and Terminology WHY PDE ?? where PDE in BC on Analytic Solution: PDEs can be used to describe a wide variety of phenomena such as - sound - heat - electrostatics -……. - electrodynamics - fluid flow - elasticity These phenomena can be formalised in terms of PDEs

13. Definitions and Terminology WHY PDE ?? PDEs can be used to describe a wide variety of phenomena such as - sound - heat - electrostatics -……. - electrodynamics - fluid flow - elasticity These phenomena can be formalised in terms of PDEs Heat equation the wave equation Laplace's equation Helmholtz equation Schrödinger equation Navier–Stokes equations Darcy law Biharmonic equation -------------

14. Definitions and Terminology Numerical methods to solve PDEs Numerical solution is almost the only method that can be used for getting information about the solution The three most widely used numerical methods to solve PDEs are Analytical solution is not available (almost all) The finite element method (FEM), The finite volume methods (FVM) The finite difference methods (FDM).

15. Math-574 Topics (Part 1) No classesTOPICS 2 Introduction and fundamental concepts. Classification of second -order linear PDE Quick introduction to PDE toolbox 5 The Finite Element Method for Poisson’s Equation Triangulations Data Storage Structures Mesh Generation The Space of Piecewise Linear Quadrature and Numerical Integration Green’s Formula Variational Formulation Finite Element Approximation Derivation of a Linear System of Equations Properties of the Stiffness Matrix Computer Implementation Assembly of the Stiffness Matrix Assembling the Boundary Conditions A Finite Element Solver for Poisson’s Equation A Priori Error Estimates 4 The Finite Element Method for Time-dependent Problems (The Heat Equation) Finite Difference Methods for Systems of ODE The Heat Equation Variational Formulation Spatial Discretization Time Discretization Computer Implementation Stability Estimates A Priori Error Estimates 3 The Finite Element Method for Time-dependent Problems (The Wave Equation) The Wave Equation Variational Formulation Spatial Discretization Time Discretization Computer Implementation Stability Estimates A Priori Error Estimates 1 Iterative Methods for Large Sparse Linear Systems Direct Methods - Iterative Methods Conjugate Gradient Method (CG) Preconditioning MINRES - GMRES Jacobi’s Method - The Gauss-Seidel Method In this course, we study the analysis, implementation and application of finite element methods. The following topics are studied in this course: