Course syllabus for Partial differential equations, first course

A solid background in modern linear algebra and calculus in one and several variables. A solid background in Fourier analysis, especially the method of separation of variables for solving PDEs.

This course gives an introduction to the modern theory of partial differential equations (PDEs) with applications in science and engineering. It also presents an introduction to the finite element method as a general tool for numerical solution of PDEs.

• derive weak formulations of the basic initial-boundary value problems for PDEs.
• derive stability estimates for the continuous problems and predict the influence of data.
• formulate Galerkin finite element methods for PDEs and dynamical systems.
• derive error estimates using exact solution (a priori) and numerical solution (a posteriori).
• explain how the finite element method is implemented in computer code.
• improve the error estimates modifying the method or employing adaptive procedure.
• draw relevant conclusions about stability, reliability and efficiency of the methods.

Weak solutions to elliptic, parabolic, and hyperbolic partial differential equations (PDE). Computation of approximate solutions to various PDE by the finite element method (as well as dynamical systems). Interpolation, quadrature and linear systems. A brief introduction to representation theorems and abstract theory to justify the weak (variational) approach. A priori and a posteriori error estimates. Applications to e.g. diffusion, heat conduction, and wave propagation. More precisely the course covers following topics: Basic interpolation theory: Interpolation with polynomials Interpolations error analys quadrature rules and quadrature error. Numerical linear algebra: Solving linear system of equation with Jacobi's method Gauss- Seidel and Overrelaxation methods. Dynamical system: Structures in approximation with polynomials Ill-conditioned system. Finite element method for boundary-value problem in 1D: Stability Error estimates and algorithms Finite element method för initial-value problem i 1D: Fundamental solution Stability Error estimates and algorithms The dual problem. Lax-Milgram theorem: Abstract formulation Riesz representation theorem Studies and Analysis of problems in higher space dimensions: Finite element in higher space dimensions. Finite element method for Poisson equation in higher space dimensions. Finite element method for heat equation in higher space dimensions. Stability Error estimates and numerical algorithms Finite element method for wave equation in higher space dimensions. Fundamental solution Stability Error estimates and numerical algorithms Finite element method for convection-diffusion equations: Stability Error estimates and numerical algorithms

Lectures (about 35 hours), exercises (about 21 hours) and home assignments consisting of both theoretical and computer assingments.

M. Asadzadeh : An Introduction to Finite Element Methods (FEM) for Differential Equations. (Available in Cremona)

M. Asadzadeh, Lecture Notes in PDE (electronic)

Home and computer assignments combined with written exam