Advanced Numerical Analysis: Finite Element Methods

Advanced Numerical Analysis: Finite Element Methods
G22.2945, G63.2040
Fall 2001, Wednesdays 5:10-7:00pm, WWH 813

Instructor: Olof B. Widlund

  • Coordinates
    Office: WWH 712
    Telephone: 998-3110
    Office Hours: Drop by any time, or send email or call for appointment
    Email: widlund@cs.nyu.edu

  • Homework
  • Lectures
    1. September 5. A finite element method for a one-dimensional problem. The difference between Dirichlet and Neumann boundary conditions. Green's formula and a variational formulation of Poisson's equation. Examples of continuous, piecewise polynomial finite element methods.
    2. September 12. NYU closed. A make-up class will, if possible, be scheduled at the end of the semester.
    3. September 19. H^1-conforming finite elements must be continuous. An example of a harmonic function with nice boundary values and a singularity at a reentrant corner; the region is non convex and does not have a smooth boundary. Friedrichs' and Poincar\'{e}'s inequalities. Bounds from below of the first and second eigenvalues of the Laplacian with Dirichlet (or mixed) and Neumann boundary conditions, respectively. Lipschitz regions and a trace theorem for H^1 on such regions.
    4. September 27. Rellish's theorem and generalized Poincar\'{e}'s inequalities. Approximation theory for finite element spaces. Examples of Lagrange finite elements, Hermite finite elements on triangles. Finite element spaces on quadrilaterals. C^1 finite elements due to Agryris, Bell, and Hsieh, Clough, and Tocher.
    5. October 3. The Nitsche-Aubin error estimate. Regularity of elliptic problems on regions with smooth boundaries or on convex regions. An example of problem on a non-convex region with limited regularity for smooth data. Error bounds for functions with limited smoothness by quasi-interpolants. Aspects of implementation of finite element methods.
    6. October 10. Why nonconforming finite element problems? Strang's Lemmas. Analysis of nonconforming P_1 elements. Why such elements? Stokes' equations. Saddle point problems.
    7. October 17. Morley's element for the biharmonic problem. L_2-bounds for the error of nonconforming finite elements. Applications to the non-conforming P_1 elements. Methods for regions with curved boundaries. Gauss-Lobatto quadrature. Isoparametric elements. Dual norms and the inf-sup condition.
    8. October 24. inf-sup stability for the continuous Stokes problem. Mixed finite element methods for Darcy's law. Norm of the solution operator for saddle point problems expressed in terms of the inf-sup parameter of the discrete mixed method. Error bounds for mixed methods in terms of the inf-sup parameter and best approximation results.
    9. October 31. Saddle point problems with a penalty term with an example from almost incompressible elasticity. An example where the bilinear form a(.,.) is only elliptic on the subspace for which the constraints are satisfied; cf. the minimal requirements in Brezzi's theorem on p. 132. Fortin interpolation. Examples of inf-sup stable and inf-sup unstable methods for Stokes equations. Tools for proving inf-sup stability of methods for Stokes equations. The stability of the lowest order Taylor-Hood elements following a paper by Rolf Stenberg.
    10. November 7. Examples of inf-sup stable finite element methods for Stokes equations. Proof to inf-sup stability for the mini element. Non-conforming P_1 elements and a divergence free basis for two dimensions. Introduction to domain decomposition: transmission conditions, the Neumann-Dirichlet algorithm.
    11. November 14. Schur complements, trace and extension theorems for finite elments. Convergence rates of the Neumann-Dirichlet algorithm. The Neumann- Neumann algorithm. The case of many subdomains. Introduction of the Schwarz alternating method.
    12. November 21. Convergence of the two subdomain multiplicative Schwarz method using arguments about Schur complements. The case of three and more subdomains. The block-Jacobi preconditioner. Using overlap to enhance the convergence. Additive Schwarz methods. Introduction to a theory for Schwarz methods based on three assumptions.
    13. November 28. Abstract theory for Schwarz methods. Yserentant's hierarchical basis method.
    14. December 5. More about Yserentant's method. Multigrid and BPX methods. Two-level overlapping Schwarz methods.
  • Required Text
    Finite Elements. Theory, Fast Solvers, and Applications in Solid Mechanics by Dietrich Braess. Cambridge University Press. Second Edition.

  • Other Good Books
    The Mathematical Theory of Finite Elements by Susanne Brenner and Ridgway Scott. 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.

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