## Numerical Methods for Partial Differential Equations

Code no:

**9.2.29.9.2.9**(9181) Semester:**8th,**Teaching hours: 4*Introductory example*

*:*Dirichlet problem. Weak form. Arithmetic solution with Finite Elements Method.

*Boundary Value Problems and*

*Galerkin method*

*:*General week form. Lax-Milgram theorem. Galerkin method. Errors. Rayleigh-Ritz-Galerkin method. General derivatives and Sobolev spaces. Green formulas. Elliptic boundary value problems. Existence and uniqueness. Applications.

*Finite*

*Elements*

*Methods*

*for*

*Elliptic*

*Boundary*

*Value*

*Problems*

*:*Piecewise polynomial, Hermite and spline interpolation. Error estimates. Applications.

*Finite*

*Elements*

*Methods*

*for*

*Evolutionary*

*Boundary*

*Value*

*Problems*

*:*Parabolic and hyperbolic problems. Euler and Crank-Nicholson methods. Stability. Error estimate. Applications.

*Finite Difference Methods*

*:*Sturm-Liouville and Dirichlet problems. Heat equation. Wave equation. Stability and Convergence.

*Workshop Exercises*

*:*Use of software (FORTRAN - IMSL, Matlab, Mathematica, Programming Libraries etc.) for programming.

specific knowledge:

- The course provides the basic background for the finite element and finite difference methods that are applied to the numerical solution of partial differential equations.
- This course aims at presenting and analysing the properties of basic numerical methods for solving boundary value problems.

skills and competences:

Upon successful completion of the course the student will be able to:

- Understand the concept of the weak solution, the discrete weak solution, the finite elements, as well as the qualitative characteristics of the finite element method (stability, linear system solvability and error estimates).
- Understand the basic methods of finite differences for the study of numerical methods of boundary value problems, as well as the qualitative characteristics of these methods (stability, linear system solvability, error estimates).
- work with her/his colleagues to solve complex practical problems.