Generally the idea is to first ask you for a story of the following kind: I have problem X (e.g. Robin boundary value problem for heat conduction in 2D) and I want to solve this numerically using the finite volume method. What do I do? After your answer on that some additional questions might follow.
Here follows the list of topics I might ask about.
Basic principles of the von Neumann Architecture for sequential processors
Julia: Multiple dispatch,Forward mode automatic differentiation
Computer representation of floating point numbers
Gaussian elimination, LU factorization, partial pivoting
Tridiagonal matrix algorithm
Sparse matrix storage, sparse direct solvers
Basic iterative methods, simple preconditioners
Sufficient condition for convergence of iterative methods
Perron-Frobenius theorem
Jacobi iteration convergence, M-Matrix criterion
Regular splittings, M-Matrices, convergence of iterations based on regular splittings
Gershgorin circles, Taussky theorem, irreducible matrices
Matrix theory: (irreducibly) diagonally dominant matrices, nonsingularity criterion,
Incomplete LU factorizations
The method of conjugate gradients
Basic idea of Krylov subspace methods
Delaunay triangulation, Voronoi diagram
Voronoi finite volume discretization, solvability of the discrete problem
Stiffness matrix assembly for FVM
Implementation of Dirichlet boundary conditions (Penalty method, elimination)
Time dependent problems, implicit/explicit Euler mehod
Convection-diffusion problem, upwinding, exponential fitting
Nonlinear diffusion, finite volume discretization, Newton method