# TMA4220 Numerical Solution of Partial Differential Equations Using Element Methods -- Autumn 2018

 Lecturer: Aurélien Larcher Maxime Conjard Monday 10–12 and Tuesday 10–12 @ L10 Friday 12–14 @ S22 *but* Computer Room Nullrommet from Week 37 onwards

Lecture and Exercise material will be upoaded exclusively to this page; only Project assignments and announcements will be communicated through Blackboard.

*Lecture on 2018-10-09 is replaced by a project guidance session regarding to assembly of finite element contributions. Computer session on 2018-10-12 is cancelled.*

*Lectures Week 44 on 2018-10-29 and 2018-10-30 are cancelled*

## Objective

The course provides an introduction to solving PDEs using finite element methods. Topics covered are: variational and weak formulations, boundary conditions, finite element spaces, integration rules, stability and error analysis, implementation, direct and iterative solvers for the resulting linear systems, application to the Poisson equation and the convection–diffusion equation.

Learning outcomes:

1. Knowledge: strong and weak formulations of linear partial differential equations, mathematical theory of finite element methods.

2. Skills: formulation and implementation of numerical solvers based on finite element methods for initial and boundary value problems in different dimensions of space, and evaluation of the quality of obtained numerical results.

3. General competence: written and oral presentation of scientific issues and results.

## Evaluation

 Project 35% Project Written examination 65% 2018-12-07 9–13

Support material code for the examination: ' C: Specified printed and hand-written support material is allowed. A specific basic calculator is allowed. '

Allowed material consists of typed Lecture Notes and hand-written notes.

The evaluation will cover the following content:

• Chapters 1 to 6 of the lecture notes (Weeks 35-41),
• the a posteriori error analysis for Poisson Chapter 7 (Week 42),
• and Exercise on stability analysis for a 1d Advection–Diffusion problem Chapter 8 (Week 43).

Requirements are:

• derivation of a weak formulation to a PDE,
• a priori stability estimate,
• use of Lax-Milgram to prove well-posedness,
• finite element framework: reference element, transport/mapping, contributions, lifting, boundary conditions,
• a priori error estimates
• stability estimate for time-dependent problems,
• residual-based a posteriori error estimate,
• stability properties for convection-dominated problems.

 A > 89 B > 77 C > 65 D > 53 E > 41

Examination of previous years:

## Lecture Plan

Lectures will be given exclusively in English.

Aside from the Quarteroni book, short notes are published weekly to summarize the content of the lectures: Short Notes

Week Subject Lecture Notes Material
34 Presentation of the Curriculum - PDF
- - -
35 Weak formulation of PDEs 1 PDF Quarteroni 3.1-4.
Weak formulation of PDEs 2 - Quarteroni 3.1-4.
36 Ritz and Galerkin methods PDF Quarteroni 3.3
Ritz and Galerkin methods - Quarteroni 4.2
37 Galerkin FEM and well-posedness PDF Quarteroni 4.3-4.4
Galerkin FEM and well-posedness (no lecture, to be continued week 38) - Quarteroni 4.5
38 Finite Element spaces PDF Ern-Guermond Chap 1-2
Finite Element spaces - -
39 Polynomial approximation and error analysis PDF Quarteroni 4.3
Polynomial approximation and error analysis - -
40 Time-dependent problems - Quarteroni 5.1
Time-dependent problems - -
41 Implementation of FEMs - Quarteroni 4.4-4.5
Implementation of FEMs Project Guidance in Nullrommet -
42 Mesh generation and adaptivity - Quarteroni 4.6, 6.
Mesh generation and adaptivity - -
43 Stabilized finite elements - Quarteroni 12.
Stabilized finite elements - -
44 No Lecture - -
No Lecture -
45 Iterative solvers and Multigrid - Quarteroni 7.
Project Guidance in Nullrommet Project Guidance in Nullrommet -
46 Mixed problems - Own notes only.
Project Guidance in Nullrommet Project Guidance in Nullrommet -
47 Examination repetition - -
Project presentation - -

## Exercises

Exercises are provided at the beginning of session in printed form together with a short note summarizing the previous lecture, and sent by email simulatenously. Suggested solutions are published the following week.

Week Subject Problem Set Solutions
35 Weak formulation Exercise Chapter 1 -
36 Galerkin approximation Exercise Chapter 2 -
37 Lagrange P1 1D Poisson Exercise Chapter 3 -
38 Lagrange P1 1D Helmoltz PDF Chapter 1-4 + EX -
39 Lagrange P2 1D Helmoltz Continue EX4 from last week. -
40 Project I Description -
41 Guidance Project I Moved to 2018-10-09 -
42 Guidance Project I -
43 Guidance Project I -
44 Project II Description -
45 Guidance Project II -
46 Guidance Project II -
47 Presentation Project II -