# Numerical Solution of Partial Differential Equations (ΑΑ6)

### General

School School of Science Department of Mathematics Graduate AA6 2 Numerical Solution of Partial Differential Equations Lectures (Weekly Teaching Hours: 3, Credits: 7.5) Special background, skills development. - Greek Yes (in English) See eCourse, the Learning Management System maintained by the University of Ioannina.

### Learning Outcomes

Learning outcomes Upon successful completion of this course, students will be able to: apply advanced numerical analysis techniques to prove error estimates for numerical approximations of elliptic and parabolic problems. demonstrate independence in the use of research materials to prove key results. write FEM code in FEniCS or Octave and construct appropriate numerical experiments to verify theoretical results. evaluate the correctness of numerical results by comparing them with both the theory of numerical methods and the theory of continuous problems. Search for, analysis and synthesis of data and information, with the use of the necessary technology. Adapting to new situations. Working independently. Decision-making. Production of free, creative, and inductive thinking. Promotion of analytical and synthetic thinking. Working in an interdisciplinary environment.

### Syllabus

• Hilbert spaces, Riesz’s representation theorem, Lax-Milgram’s theorem, Cea’s theorem.
• Sobolev spaces, weak derivatives, Poincare-Friedrichs inequalities.
• Weak formulation and the Finite Element Method (FEM) for elliptic boundary value problems in 1D and 2D. A priori and a posteriori error estimates, adaptivity.
• Semi-discrete and fully-discrete schemes for parabolic equations. Temporal discretization with the Explicit and Implicit Euler methods, and the Crank-Nicolson method.
• Computer implementation of FEMs.

### 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 learning platform.
• Provision of study materials in Moodle e-learning platform.
• Use of online quizzes in Moodle platform, which aim to enhance student engagement and motivation in learning.
• 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 (Octave ή FEniCS) for the computer implementation of FEM.
Teaching Methods
Lectures 39
Study and analysis of bibliography 70
Worksheets 25
Project 35
Presentation 18.5
Course total 187.5
Student Performance Evaluation
• Solution of worksheets (Weighting 30%, addressing learning outcomes 1-2)
• Project (Weighting 45%, addressing learning outcomes 1-4)
• Presentation (Weighting 25%, addressing learning outcomes 1-5)

### Attached Bibliography

• “Μέθοδοι πεπερασμένων στοιχείων”, Γ. Δ. Ακρίβης, Λευκωσία, 2005.
• “Αριθμητική λύση μερικών διαφορικών εξισώσεων”, Μ. Πλεξουσάκης, & Π. Χατζηπαντελίδης, Κάλλιππος, 2015. http://hdl.handle.net/11419/665
• “The Mathematical Theory of Finite Element Methods”, S.C. Brenner, & L.R. Scott (Third ed., Vol. 15), Springer, New York, 2008.
• “Galerkin Finite Element Methods for Parabolic Problems”, V. Thomee, Springer-Verlag, 1997.