Advanced Numerical Analysis: Finite Element Methods

G22.2945.001, G63.2012.001
Spring 2011, Mondays 1:25 - 3:15 pm, WWH 517

Instructor: Olof B. Widlund

  • Coordinates
    Office: WWH 612
    Telephone: 998-3110
    Office Hours: Mondays 11:30am - 12:30pm or drop by any time, or send email or call for an appointment.

  • Main Text
    Finite Elements. Theory, Fast Solvers, and Applications in Solid Mechanics by Dietrich Braess. Cambridge University Press. Third Edition.

  • Other Good Books
    The Mathematical Theory of Finite Elements by Susanne Brenner and Ridgway Scott. Third Edition. Springer.
    Numerical Analysis of the Finite Element Method by Philippe Ciarlet. University of Montreal Press.
    Numerical Solution of Partial Differential Equations by the Finite Element Method by Claes Johnson. Cambridge University Press.

  • Final Exam
    This is a seminar course. Any student, who so desires, can sign up for an individual oral exam at the end of the term or some time in Summer 2011. Please send e-mail requesting a date and time.

  • Lectures These are short descriptions of the content of each lecture, after the fact. Note that the 2/7/11 class was canceled and will be replaced by a lecture on May 16. 2/21/11 was an NYU holiday and 3/14/11 falls during the NYU Spring break week.
    1. January 24. A discussion of text books. Poisson's problem; a simple scalar elliptic problem in operator form. Green's formula and how to convert this problem into variational form. Dirichlet, Neumann, and mixed boundary conditions. A few words on the Sobolev space H^1. Ellipticity and boundedness of the bilinear form. Finding the same solution by solving a minimization problem. Triangulation of domains in two and three dimensions. Standard Lagrangian and some Hermitian elements. Conforming finite element spaces; for second order problem continuity is necessary. H^2 conforming finite elements: they are all quite complicated, e.g., Agryris and Bell elements and the Hsieh- Clough-Tocher elements. The Bogner-Fox-Schmit element for rectangles.
    2. January 31. Cea's lemma and the issues that arise in its application. The Aubin-Nitsche result; a different issue on the regularity of solutions arises, which requires a smooth boundary of the domain or a convex domain. An example of a problem which does not give the full regularity required in the Aubin-Nitsche context. The definition of W^k_p spaces for positive k, not necessarily integer; k=1/2 needed when discussing the traces of elements in H^{1/2}. The need of more than minimal regularity in defining the standard, local finite element interpolants.
    3. February 7. Class canceled; the instructor in San Diego attending a domain decomposition conference. A make-up class is scheduled for May 16.
    4. February 14. The structure of the finite element problems and and the standard nodal bases an how this is all reflected in the sparsity and other properties of the stiffness and mass matrices. How to use Friedrichs inequality and Gershgorin's theorem to estimate the condition number of the stiffness matrices. Poincare's and Friedrichs inequalities; they can be proven for simple geometries by using calculus. Rellish theorem and more general Poincare-Friedrichs inequalities.
    5. February 21. NYU holiday.
    6. February 28. A generalized Poincare inequality expressed by using the H^k-norm of a quotient space. How different terms of a full H^k-norm scale under a dilation. Using a Sobolev inequality to estimate the norm of finite element interpolants by higher order H^k norms. Bramble-Hilbert estimates which combined with Cea's or Aubin-Nitsche's result give the basic error bound for conforming finite elements. The need to consider non-conforming finite element spaces and the need to modify the bilinear forms and the functionals in the right hand side, e.g., when numerical quadrature must be used. Proofs of Strang's first and second lemmas. Application of the second to the case of the discontinuous P_1 elements also know as the Crouzeix-Raviart elements.
    7. March 7. Lipschitz domains and trace spaces and estimates for H^1. A problem on domains which are not the union of triangles: an application of a Strang lemma. Isoparametric elements; a main issue is if the Jacobian of the mapping vanishes. Two cases considered namely that of general quadrilaterals, using Q_1 functions to build the mapping and the finite element functions on the reference square and the case of using P_2 for triangles with one side a parabola. Model problems for mixed finite element methods: flow in porous media, incompressible Stokes, and almost incompressible elasticity; to be continued.
    8. March 14. NYU Spring break.
    9. March 21. Almost incompressible elasticity solved by mixed finite element methods. Saddle point systems of linear algebraic equations and explicit formulas for their solution. Bounds for the energy of the different components of the mixed system in terms of dual norms of the right hand sides. The inf-sup condition and how the inf-sup parameter affects these bounds. The solution of a related continuous problem div u = p, with and without Dirichlet boundary conditions for the vector valued solution u. How to derive error bounds for the mixed system using the bounds obtained by linear algebra tools; a small inf-sup parameter will lead to loss of accuracy. Another look at the second model problem for flow in porous media; in one case we need to consider problems posed in H(div), a space intermediate between H^1 and L_2.
    10. March 28. Ellipticity of the first bilinear form of the elasticity operator; Korn's first inequality, without a proof, and the second, proven by using the first and Rellish's theorem. Derivation of the rigid body modes, which span the null space of the whole elasticity operator. Flow in porous media and two mixed finite element approaches. The need for a modification of the bounds derived last week when we work with the space H(div). Raviart-Thomas element, which are conforming in H(div). inf-sup stability of a pair of spaces, one of which is the lowest order Raviart-Thomas space. A comment on the Arnold-Brezzi use of Lagrange multipliers and local elimination of all other variables to return to a positive definite problem. Examples of a unstable pair of finite element spaces for incompressible Stokes and several fixes including the use of the mini element. Using Crouzeix-Raviart elements to solve the incompressible Stokes problem and a basis for this space for the two-dimensional case.
    11. April 4. inf-sup theory for the incompressible Stokes problem following Stenberg, International Journal for Numerical Methods in Fluids, vol. 11, 1990, pp. 935-948. An introduction to domain decomposition algorithms based on slides developed for a tutorial at the 20th international conference on domain decomposition methods held in February 2011. There is also a four page introduction to this tutorial. This lecture covered the first 22 of the 98 slides.
    12. April 11. Continuation of the discussion of domain decomposition algorithms using the slides of the tutorial starting with a short review of what was done last week and then covering slides 23 to 50.
    13. April 18. Continuation of the discussion of domain decomposition algorithms using the slide set, essentially, slides 51-69 this week.
    14. April 25. Continuation of the discussion of domain decomposition algorithms using the slide set, essentially, slide 70 to the last one. The main topics were recent overlapping Schwarz methods for almost incompressible elasticity and basic theory for FETI-DP and BDDC algorithms.
    15. May 2. More about FETI-DP and BDDC; the connection to analysis of other domain decomposition algorithms. An introduction to multi-grid methods. The hierarchical basis method of Yserentant.
    16. May 9. More on Yserentant's algorithm. Zhang's proof of the optimality of the multigrid V-cycle and related methods. A posteriori error estimates, in particular, those based on residuals.
    17. May 16. Make-up class. H(div) problems. Raviart-Thomas element spaces for arbitrary values of k. Continuity of the normal component guarantees that these finite element are in H(div). H(curl) problems and Nedelec elements in 2D; they can be obtained by a rotation from the Raviart-Thomas elements. H(curl) and Nedelec elements in 3D. The most important degrees of freedom are then associated with edges of the elements. The Nedelec elements are H(curl)-conforming since they have continuous tangential components. A few additional comments on a posteriori error estimates.