### Spring 2013: Computational Methods for Geophysics (GEO 384F)   This course treats numerical methods for the solution of partial differential equations arising in continuum geophysics. Our focus is on the finite element method (FEM), for its generality, adaptivity, and accuracy. We will develop the core ingredients of the FEM - weak formulation, Galerkin approximation, piecewise polynomial basis functions, numerical quadrature, isoparametric elements, assembly - with reference to a model potential problem. While the FEM method is applicable to a broad spectrum of geophysical models including those arising in meteorology, climate, seismology, geodynamics, subsurface flow, etc., we will consider a subset of problems depending on the interests of the students. Past settings have included heat conduction and viscous flow in mantle convection and ice sheet dynamics, seismic wave propagation, and porous media flow and transport. We will use a high-level finite element toolkit (FEniCS) to build simulators in each of these areas and Paraview for visualisation. [flyer]
 Instructor: Omar Ghattas Office: ACE 4.236A Phone: +1 512.232.4304 E-mail: omar at ices.utexas.edu Web: http://users.ices.utexas.edu/~omar Teaching Fellow: Noémi Petra Office: ACE 4.252 Phone: +1 512.232.3453 E-mail: noemi at ices.utexas.edu Web: http://users.ices.utexas.edu/~noemi

Lectures: Monday and Wednesday, 10am - 12noon, JGB 2.202

Office hours: Monday and Wednesday, 12noon - 1pm, ACE 4.252 (in ICES)

#### Class overview and topics:

1. Jan 14 (M): Overview and introduction
2. Jan 16 (W): Discussed a basic finite difference method which we applied to solve the Poisson problem; discussed the approximation error and compared the true and approximation solutions. Energy formulation, calculus of variation.
3. Jan 21 (M): Martin Luther King Jr. Day holiday.
4. Jan 23 (W): Continued the Energy formulation, calculus of variation. The Ritz method. [1st assignment]
5. Jan 28 (M): We continued the Ritz method; looked at an example, computed the error in inf, L2 and energy norms; showed that the Ritz method minimizes the error when measured in the energy norm.
6. Jan 30 (W): The Galerkin method.
7. Feb 04 (M): Example: Finite element basis functions and calculations. [2nd assignment]
8. Feb 06 (W): Assembling the stiffness matrix and load vector (in local coordinate system using shape functions); we also learned how to handle boundary conditions.
9. Feb 11 (M): Error analysis.
10. Feb 13 (W): Error analysis (continued with numerical demostration) [3rd assignment] ; Matlab files used in class can be found [here].
11. Feb 18 (M): Introduction to FEniCS [slides, additional notes, code]
12. Feb 20 (W): Continued with FEniCS demostration for the convergence analysis discussed previously; we also learned how to set up mixed (and inhomogenous) boundary conditions. [4th assignment] ; FEniCS files used in class can be found [here].
13. March 04 (M): We concluded 1d problems (i.e., discussed more general BCs, more general equations, and higher order elements).
14. March 06 (W): We finished the discussion on higher order elements; we also had a FEniCS session in which we reviewed the 1d codes used/extended for HW4.
15. March 11 (M): Spring break: [Have a great vacation!!!]
16. March 13 (W): Spring break: [Have a great vacation!!!]
17. March 18 (M): Two-dimensional problems (see Chapter 4 in the text). We considered the Poisson equation in 2D as an example problem, talked about conservation of energy, the divergence theorem, the Green's first identity, and the finite element approximation.
18. March 20 (W): The weak form of the Poisson equation in 2D and its finite element discretization. Piecewise-linear interpolation on triangles.
19. March 25 (M): Quadratic triangle, higher order triangular elements, rectangular finite elements [5th assignment]
20. March 27 (W): An example of linear triangles (Poisson 2D problem with Dirichlet, Neumann and mixed boundary conditions.)
21. April 01 (M): FEniCS session: We solved the Poisson 2D problem with Dirichlet and mixed boundary conditions, looked at convergence rates for linear and quadratic finite elements and introduced ParaView as visualization tool. The FEniCS files used in class can be found [here].
22. April 03 (W): Finite elements for linear elasticity (displacement form of the linear elasticity boundary value problem (strong form))[6th assignment]
23. April 08 (M): Finite elements for linear elasticity (component form and discretization)
24. April 10 (W): A simple example of FEM for elasticity
25. April 15 (M): Nonlinear problems (nonlinear Poisson, Newton's method at the discrete level)
26. April 17 (W): Nonlinear Poisson (Newton's method in variational form) [mini-project] (note the time change)
27. April 22 (M): FEniCS session: We solved a linear elasticity problem, the nonlinear Poisson problem (with both the fenics built in nl solver and with a manually written Newton solver), looked at fem convergence rates for a nl problem with manufactured solution, also showed how to handle more complex geometries. The FEniCS files used in class can be found [here].
28. April 24 (W): Stokes flow (strong and weak forms and discretization)
29. April 29 (M): Nonlinear Stokes coupled with the energy equation (modeling for ex. mantle convection, ice sheet dynamics) and the Navier-Stokes equation
30. May 01 (W): Initial-boundaty value problems (e.g., heat equation, time discretizations, example Crank-Nicholson). We also had a FEniCS session in which we showed how to solve a time dependent problem and a coupled problem (Stokes + adv-diff eq).The FEniCS files used in class can be found [here].