Numerical Solution of Partial Differential Equations (MAE882)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAE881
Semester	8
Course Title	Numerical Solution of Partial Differential Equations
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 6)
Course Type	Special background, skills development.
Prerequisite Courses	-
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes

Upon successful completion of the course, students will be able to: describe and apply key numerical methods for the solution of boundary/initial value problems for elliptic and parabolic equations (e.g., Laplace, Heat Equation). be aware of the optimal order of accuracy of key numerical methods as well as the limitations that may be required in the discretization parameters to ensure stability. apply common techniques for analyzing finite difference and finite element methods. implement finite difference and finite element methods (in Octave ἠ Python) to compute the numerical solution of PDEs and calculate their experimental order of convergence.

General Competences

Search for, analysis and synthesis of data and information, with the use of the necessary technology. Adapting to new situations. Decision-making. Working Independently. Production of free, creative, and inductive thinking. Promotion of analytical and synthetic thinking.

Syllabus

• Finite difference approximations to derivatives.
• The two-point boundary value problem. Boundary conditions of type Dirichlet, Neumann, and Robin.
• Finite differences schemes for the two-point boundary value problem. Consistency and stability. The energy method. Order of accuracy and convergence.
• The Finite Element Method (FEM) for the two-point boundary value problem. A priori and a posteriori estimates. Implementation of FEM.
• Finite differences and Finite element methods for the Heat Equation in 1D. Explicit- and implicit Euler, the Crank-Nicolson method. Consistency and stability.
• The finite element method for elliptic and parabolic equations in higher dimensions.

Teaching and Learning Methods - Evaluation

Delivery	Face-to-face.
Use of Information and Communications Technology	Use of a tablet device to deliver teaching. Lecture materials in pdf-format are made available to students, for later review, on Moodle e-learning platform. Provision of study materials in Moodle e-learning platform. Provision of model solutions for some exercises in podcast format. Communication with students through e-mails, Moodle platform and Microsoft Teams. Use of sophisticated software (Python or Octave) for the implementation of the numerical algorithms.
Teaching Methods	Activity Semester Workload Lectures 39 Study and analysis of bibliography 75 Directed study of Computer-based exercises 6 Non-directed study of Computer-based exercises 30 Course total 150
Student Performance Evaluation	Computer-based exercises with oral examination (Weighting 35%, addressing learning outcomes 2 and 4) Written examination (Weighting 65%, addressing learning outcomes 1-3)

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• “Αριθμητική επίλυση μερικών διαφορικών εξισώσεων”, Π. Χατζηπαντελίδης, & Μ. Πλεξουσάκης, Κάλλιπος, (2015). http://hdl.handle.net/11419/665
• “Αριθμητικές Μέθοδοι για Συνήθεις Διαφορικές Εξισώσεις”, Γ. Δ. Ακρίβης, & Β. Α. Δουγαλής., Πανεπιστημιακές Εκδόσεις Κρήτης, Ηράκλειο, 2η έκδοση, 2013.
• “The mathematical theory of finite element methods”, S. C. Brenner & L. R. Scott (Third ed., Vol. 15), Springer, New York, 2008.
• “Partial differential equations with numerical methods”, S. Larsson, & V. Thomée, Springer-Verlag, Berlin, 2009.