Numerical Analysis of Partial Differential Equations
Code
11628
Academic unit
Faculdade de Ciências e Tecnologia
Department
Departamento de Matemática
Credits
6.0
Teacher in charge
Oleksiy Karlovych
Teaching language
Português
Objectives
At the end of this course students will have acquired basic knowledge and skills in the area of numerical analysis of partial differential equations of in order to:
- Understand advanced contents in the area (including Finite Difference Methods for elliptic, parabolic and hyperbolic problems, as well as the finite element method for elliptic problems);
- Being able to start research on a topic in the area.
Subject matter
1. Green''s function. Finite Difference Method for elliptic problems. Finite Difference Method for parabolic and hyperbolic problems. Explicit and implicit methods. Crank-Nicholson method. Stability. Convergence.
2. Finite Element Method for elliptic problems. Bases, matrix and global vectors. Calculation of the solution and its derivative. Convergence order of the derivative at different points of each interval of the partition. Finite element. Local bases. Matrices and vectors of the element. Global matrices and vectors. Boundary conditions of Dirichlet, Neumann and mixed type. Quadratic and cubic elements.
3. Finite element method for two-dimensional problems. Triangular and rectangular finite elements. Matrices and vectors of the element. Global matrices and vectors. Reference finite element.
4. Classic error estimates.
5. Error estimates for the finite element method for elliptic problems in Hilbert spaces. Aubin-Nitsche’s method.
Bibliography
P.G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978.
D. Euvrard, Résolution Numérique des Équations aux Dérivées Partielles, 2e edition, Masson, Paris, 1990.
T.J.R. Hughes, The Finite Element Method: Linear, Static and Dynamic Finite Element Analysis, Prentice - Hall, Englewood Cliffs, 1987.
C. Johnson, Numerical Solution of Partial Differential Equations by the Finite Element Method, Cambridge University Press, Cambridge, 1992.
J.T. Oden and G.F. Carey, Finite Elements I: An Introduction (with E.B. Becker); II: A Second Course; III: Computational Aspects; IV: Mathematical Aspects; V: Special Problems in Solid Mechanics; VI: Special Problems in Fluid Mechanics, Prentice Hall, Englewood Cliffs, 1981--1984.