# PCS912 Advanced Computing with Domain Decomposition Methods

## Course description for academic year 2024/2025

### Contents and structure

Scientific and engineering problems are often described in terms of partial differential equations which are solved numerically. Discretization methods used for the numerical solution often give rise to enormously large systems of equations. Standard techniques to solve such systems are in many cases highly expensive in terms of computation, and may even fail due to the structure of the problem itself, e.g., its coefficients and its nonlinearity. This means that the use of efficient techniques such as domain decomposition methods is almost inevitable. The design and application of such methods require advanced knowledge and skills. The purpose of this course is to study the core of advanced methods within domain decomposition and their application in engineering computing.

The course will include: numerical solution of partial differential equations based on the finite element discretization; basic iterative methods based on Krylov iterations and preconditioning; the domain decomposition concepts: Schwarz and Schur; Abstract Schwarz framework; Additive Schwarz Method, Neumann-Neumann, Finite Element Tearing and Interconnect (FETI), FETI Domain Decomposition (FETI-DP); convergence analysis; multiscale domain decomposition based on coarse enrichment; nonlinear domain decomposition based on subspace correction.

### Learning Outcome

Upon completion of the course the student should be able to:

Knowledge

• explain the general construction of domain decomposition methods and their classification, for linear partial differential equations, e.g. linear elliptic PDEs.
• explain the general construction of domain decomposition methods for central nonlinear partial differential equations, e.g. nonlinear elliptic PDEs.
• describe the principles underlying domain decomposition methods for multiscale problems.

Skills

• derive domain decomposition methods for linear and nonlinear problems.
• conduct convergence analysis of the methods for linear, multiscale, and non-linear problems.
• evaluate the algorithmic aspects and parallel performances of domain decomposition methods.
• implement and apply domain decomposition methods to a given elliptic partial differential equation using MatLab or Python.
• interpret and discuss the results obtained from application of domain decomposition methods to elliptic partial differential equation.

General competence

• classify the principles behind the construction of a domain decomposition method
• assess the applicability of domain decomposition methods to practically solving engineering problems in a well-structured manner.
• discuss and present state-of-the art in research within the theory and application of domain decomposition methods.

### Entry requirements

General admission criteria for the PhD programme.

### Recommended previous knowledge

PCS911 Engineering Computing or equivalent, background in (numerical) linear algebra, partial differential equations (PDEs), basic numerical computing, and familiarity with parallel computing.

### Teaching methods

The course consists of a combination of lectures and seminars. The lectures will be used for covering the core material of the course. Seminars permit participants to present and discuss recent research papers and for presentation by researchers in the field.

### Compulsory learning activities

There will be 2 to 3 (maximum) smaller assignments related to the lectures.

The small assignments must have been approved in order to take the exam.

### Assessment

Assignment and oral exam.

The course is graded pass / fail based on the research paper (the larger project) and an oral exam. Each of the two components must result in a pass grade in order to obtain a pass grade for the entire course.

The project work may be undertaken in groups but is required to have an individual component. Each participant must present the results from the project in the form of a research paper and presented at a seminar. The project will focus on solving a large scale computational problem in engineering (e.g., determining pressure distribution in a porous media flow), and address the computational aspect of the problem and the solution methods (e.g., the domain decomposition).

### Examination support material

Assignment (project work): All support material is permitted.

Oral exam: No support material.