Application examples of the Galerkin method with finite elements to one-dimensional two-point boundary value problems and to two-dimensional Dirichlet problems.
• Hilbert spaces, Projection Theorem.
• Boundary Value Problems and Galerkin Method: General weak form. Lax-Milgram theorem. Galerkin theorem. General error estimation.
• Generalized derivatives and Sobolev spaces. Green's formulas.
• Elliptic boundary value problems. Existence and uniqueness.
• Finite Element Methods for Elliptic Boundary Value Problems: One-dimensional finite elements. Piecewise polynomial functions. Cubic Hermite functions and splines. Two-dimensional and three-dimensional finite elements. Error estimation.
• Finite Element Methods for Evolutionary Boundary Value Problems: Parabolic and hyperbolic problems.
• Applications: Fluid flow, Heat flow, Loaded beam.
• Finite Differences: One-dimensional two-boundary value problem, Inhomogeneous Dirichlet problem for Poisson equation (5-point scheme), Heat equation (FTCS, Crank-Nicolson, Stability), Hyperbolic problems (CFL condition).
• The aim of the course is to understand the basic finite element and difference methods for solving boundary value problems, as well as the qualitative characteristics of these methods (stability, linear system solvability and error estimates).
- Teacher: Βασίλειος Κοκκίνης
ECTS : 3
Language : el
Learning Outcomes : Upon successful completion of the course, students will be able to:
• Understand the concept of weak solution, discrete weak solution, finite elements, as well as the qualitative characteristics of the finite element method (stability, linear system solvability and error estimates).
• Understand the basic finite difference methods for the numerical treatment of boundary value problems, as well as the qualitative characteristics of these methods (stability, linear system solvability and error estimates).
• Be able to collaborate with fellow students to solve complex practical problems using the above methods.