### Lecturer: Jed Brown, PhD student, VAW, ETH Zurich

### PDF version. Updates will be posted here.

#### Update 2009-01-27: slides available for lecture 2

#### Update 2009-01-29: slides available for lecture 3

#### Update 2009-01-29: References

#### Update 2009-02-05: slides available for lecture 5

### Time and place

- Five lectures: Tuesday and Thursday, 13:00 to 14:00
- Chapman building, room 106

### What is a scalable solver?

- A
*scalable solver*can solve*N*equations in*N*unknowns with*O(N)*work. - Newton-Krylov methods offer:
*Quadratic*convergence on the nonlinearity- Parallel scalability and
*mesh-independence*for the linear solve

### Is it time to look at your solver?

- Is your solver using significantly more time or memory than the physics?
- Is your time stepping limited by stability?
- Are you putting loops
*around*the analysis?

### Proposed schedule

- Jan 22: Nonlinear systems
- Motivation for coupled implicit methods
- Scalability
- Newton's method for large systems
- Globalization

- Jan 27: Linear solvers slides (pdf)
- Limitations of direct methods
- Krylov methods
- Representation of matrices and Jacobian-free methods
- Preconditioners for simple definite problems
- Parallel scalability

- Jan 29: Constraints and coupling slides (pdf)
- Preconditioners for indefinite problems
- Preconditioners for multi-physics

- Feb 3: Parallel software day: PETSc
- Generic solver components
- Physics-based preconditioners
- Working with legacy code

- Feb 5: Higher order finite elements slides (pdf)
- High-order elements at the cost of low-order elements
- Exploiting the memory hierarchy and tensor-product operations

#### Schedule and content are flexible, let me know if you have requests.

#### Have a look at my AGU poster for a bit of a preview.

### References

- Kelley.
*Solving nonlinear equations with Newton's method*, 2003. - Benzi, Golub, Liesen.
*Numerical solution of saddle point problems*, Acta Numerica, 2005. - Elman et. al.
*A taxonomy and comparison of parallel block multi-level preconditioners for the incompressible Navier-Stokes equations*, JCP, 2008. - Knoll, Keyes.
*Jacobian-free Newton-Krylov methods: a survey of approaches and applications*, JCP, 2004.

The Knoll and Keyes paper is especially recommended. Please email me if you need a copy of these or anything else.