Finite Differences and Finite Elements Method

Mode of Delivery

Face to Face

Prerequisites and Co-Requisites

None

Recommended Optional Programme Components

Numerical Analysis, Diferential Equations

Course Contents

This course equips students with in-depth knowledge on; numerical techniques, introduction to a finite difference method, a mesh generation for the finite difference method, solutions of laplace and poisson’s equations based on the finite difference method (two dimensional), an application of the finite difference method in geophysics, an application of the finite difference method on resistivity modeling and to develop a two dimensional code, three dimensional resistivity modeling based on the finite difference method, introduction to a finite element method, a mesh generation for the finite element method, solutions of Laplace and poisson’s equations based on the finite element method (two dimensional), comparison of the finite difference and finite element methods for Laplace’s equation, an application of the finite element method on resistivity modeling, how to use and run a two dimensional resistivity modeling and inversion code developed based on the finite element method,

Weekly Schedule

1) Introduction to Numerical Methods Error Types Numerical calculations and errors Errors due to express the numbers on the computer Errors arising from transactions with integer and real numbers on computers
2) Numerical Solution of Differential Equations Definition of ordinary and partial differential equations Analytical solution of ordinary differential equations Numerical solution of ordinary differential equations (Euler's Method) Partial differential equations Types of partial differential equations
3) Introduction to finite difference method Taylor series Forward, backward and central differences Error analysis
4) Mesh design in finite difference method Solution of the Laplace equation and boundary conditions Systems of equations Ax = b-type solution methods
5) The solution of Poisson's equation with finite difference method (2D) The solution of heat equation with finite difference method The solution of wave equation with finite difference method
6) Forward solution by finite difference methods in geophysical problems Development of software for direct current resistivity method and numerical solution of some others geophysical methods
7) The solution of Laplace and Poisson equations with finite difference method The soltion of Maxwell's equations by Yee's algorithm
8) Midterm examination/Assessment
9) Forward soltion for 3-D direct current resistivity method Algorithm development for the 3D-DC forwad problem
10) Introduction to finite element method Variational method Inner product space, linear operator Rayleigh-Ritz method Weighted residual method (Galerkin and other methods)
11) Mesh design in Ffnite element method Obtainig local equations in FE Obtainig global equations in FE Solution of Ax = b
12) Laplace equation and boundary conditions with finite element. Mesh desing of Laplace's equation by triangles Finite element solution of Laplace's equation
13) Poisson's equation solution with finite element. Mesh design of Poisson's equation by triangles Finite element solution of the Poisson equation
14) Comparison of finite element and finite difference methods (2B). Direct current resistivity method and inverse solution of the 2D-DC
15) Automatic mesh design in Finite element method Delaunay Triangulations Methods of mesh design
16) Final examination

Recommended or Required Reading

1- Sadıku, M.N.O, 1992, Numerical Techniques in Electromagnetics, CRC Press.
2- Heath, M. T., 1997, Scientific Computing, McGraw-Hill Press.
3- Arı, N, Özen, Ş., Çolak, Ö. H., Teşneli, A. Y., 2008, Elektromanyetik SOnlu Farklar Metodu, Palme Yayıncılık.
4- Ders notları, makaleler ve bazı programlar dersin internet adresinden indirilebilecek.
5- Lecture notes, papers and some software will be available at the internet address of the class.
6- Ödev ve çözümleri dersin internet adresinden indirilebilecek.
7- Homeworks and their solutions will be available at the internet address of the class.
8- Sınavlar ve cevapları dersin internet adresinden indirilebilecek.
9- Exams and their answers will be available at the internet address of the class.

Planned Learning Activities and Teaching Methods

1) Lecture
2) Discussion
3) Demonstration
4) Group Study
5) Problem Solving

Assessment Methods and Criteria

Contribution of Semester Studies to Course Grade	70%
Number Percentage Semester Studies Midterm Examination 1 10% Quiz 4 60% Project 1 20% Presentation/Seminar 1 10%
Contribution of Final Examination to Course Grade	30%
Total	100%

Language of Instruction

Turkish

Work Placement(s)

Not Required