Questions

Below you will find some sample question that may help you better digest the material for this course. You may find them useful preparing yourselves for the quizzes. They should be particularly useful to those who will choose Graduate Numerical Analysis as a part of their Comprehensive Exam.

• Discuss weak formulation of boundary value problems and their relation to the corresponding classical formulations
• Consider functions defined on a periodic domain. How to calculate efficiently Sobolev norms of these functions when the differentiability index is non-integer?
• What alternatives exist as regards numerical solution of boundary value problems with symmetric operators?
• What is the difference between conforming and non-conforming Finite Elements?
• How can one increase the accuracy of the numerical solution obtained with a Finite Element Method (h-refinement vs. p-refinement)?
• Consider the problem addressed in the Example "fem_01.m" posted on the course website. How would the properties of the system matrix change if sin(k*Pi*x), k=1, ..., N were used as the basis functions?
• What determines the sparsity structure of the "stiffness" matrix in the Finite Element Method
• What does it mean that a Finite Element Method is conforming?
• You are given a simple, second-order boundary value problem in 1D [e.g., -u''-u=f in (0,1)] with either Dirichlet or Neumann boundary conditions. Propose a simple Finite Element Method that you could use in order to solve this problem. Discuss how the boundary conditions in the problem will affect the solution methodology.
• Given a linear (in 1D), triangular / quadrilateral (in 2D) Finite Element, show how to contract a system of C^0/C^1 Lagrangian basis functions on this element
• What are Gauss quadratures? Discuss their properties that you think are important from the point of view of the Finite Element Method.
• Discuss how the reference element technique can be used to evaluate numerically the integrals which appear in the Finite Element Method.
• What are the properties that must be satisfied by the PDE, so that the Ritz-Galerkin method could be applied?
• Why is coercivity important in the Finite Element solution of Partial Differential Equations?
• What additional precautions need to be taken when applying a Finite Element Method to solve a first-order Partial / Ordinary Differential Equations?