## Advanced Topics in Numerical Analysis: Finite Element Methods

### MATH-GA.2012.001, CSCI-GA.2945.001 Spring 2014, Wednesdays 1:25 - 3:15 pm, WWH 312 Instructor: Olof B. Widlund

• Coordinates
Office: WWH 612
Telephone: 998-3110
Office Hours: Wednesdays 11:30am - 12:30pm or drop by any time, or send email or call for an appointment.
Email: widlund@cims.nyu.edu

• 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. SIAM.
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. Please send e-mail requesting a date and time.

• Lectures These are short descriptions of the content of each lecture, after the fact.
• January 29: A discussion of textbooks on finite elements; Braess' book, in particular its chapter 2 provides a good beginning. Poisson's equation in operator form and formulated as a variational problem in the Sobolev space H^1. Dirichlet and Neumann boundary conditions; they enter the problem in different ways. Ellipticity and boundedness in H^1 of the bilinear form that arises from using Green's formula. A few words on the triangulation of two- and three-dimensional domains. Eigenvalues of the Laplace operator with Dirichlet and Neumann boundary conditions. Poincare's inequality. Barycentric coordinates, which provide very convenient tools when describing and working with many finite element spaces. Lagrangian and Hermit elements which are conforming in H^1. A proof that H^1 conforming elements must be continuous across the interface between elements. The biharmonic equation providing a reason to consider C^1 elements. Two examples of H^2 conforming elements, one with piece-wise P^5 functions and the other using P^3 macro elements where each element is partitioned into 3. A proof that these elements are C^1 but that their basis functions cannot be written in terms of the barycentric coordinates.
• February 5: Some finite elements on quadrilaterals and hexagons. Q_k spaces. Bounds for the error in the Sobolev norm appropriate for the elliptic problem: Cea's lemma, which enables us to prove error bounds by studying the interpolation problem on individual elements. An example of an element in H^1 which is not bounded and for which the standard interpolant is not defined. Sobolev spaces including those with fractional indices. Sobolev's inequality providing bounds for the maximum of functions and derivatives of functions in terms of norms based on L_2. Rellish theorem and how to derive generalizations of Poincare's inequality. An illustration that Neumann conditions do not necessarily survive when we take a limit of a sequence of H^1 functions.
• February 12: Error bounds based on Cea's lemma and generalized Poincare inequalities: first work on an element of diameter 1 and then transform it by a trivial transformation. Aubin-Nitsche's bound for the L_2 norm of the error. That result is not valid for all elliptic problems since for second order problems a general H^2 regularity result is required. Such a result is available if the boundary of the domain is smooth or convex. An example of a problem for which H^2 regularity does not hold. A few words on how the stiffness and mass matrices of a finite element problem is computed and a few words on the condition number of stiffness matrices.
• February 19: How to get bounds for the condition number of stiffness matrices by using Gerschgorin's theorem, Rayleigh quotients and an upper and lower bounds for the eigenvalues of the mass matrix. Maximum principle; very rarely available. The clamped plate problems and the Dirichlet problem for the biharmonic. Conforming approximations requires complicated higher order elements. The Morley element which is not even conforming in H^1. The Fraeijs de Veubeke triangle and Wilson's brick; the latter two examples of finite elements defined not just by nodal values. Strang's first and second lemmas. Analysis of the discontinuous P_1 element of Crouzeiz-Raviart.
• February 26: A second application of the Strang lemmas for problems defined on domains which do not allow an exact triangulation using triangles with straight edges; this is a result for piece-wise linear elements. A few words on how to generalize this result to higher order Lagrange finite elements; the points on the curved side of the triangles next to the boundary should be selected as Gauss-Lobatto-Legendre nodes in the arc-length. The basic trace theorem for Lipschitz domains. The equations of linear elasticity for compressible and almost incompressible material. The need for alternative methods in the almost incompressible case; an additional variable, the pressure, is introduced and a saddle point problem arises quite similar to that of incompressible Stokes. An exercise in block-Gaussian elimination to discover necessary conditions for the pair of finite element spaces now required.
• March 5: Isoparametric elements, in particular using P_2 Lagrangian elements to define the mapping from the reference element to elements with one curved edge. Solving linear systems arising from mixed finite element formulations of almost incompressible elasticity. The central role of a parameter beta which appears in the denominators of expression in the right hand sides. Later it will be revealed that this is the inf-sup parameter of mixed finite element theory. A second example leading to mixed finite element problems arise in studies of flow in porous media. The H(div) space and the lowest order Raviart-Thomas elements, which are conforming in H(div). General saddle point problems arising in mixed finite elements and how to eliminate one of two right hand sides. In the continuous case this leads to a study of Necas equation.
• March 12: The hand-out of two weeks ago revisited; the parameter beta is actually the inf-sup parameter prominent in the theory of mixed finite element methods. Fortin's criterion which gives a necessary and sufficient condition for inf-sup stability of a mixed finite element pair of spaces. An unstable pair for incompressible Stokes and how it can be fixed resulting in an awkward finite element pair. Taylor-Hood families with continuous pressure spaces. The modification of P_2-continuous P_1 by partitioning the triangular elements of the velocity space into four subtriangles. The MINI element in which the velocity space is enriched by bubble functions. A return to the discussion of mixed methods for scalar elliptic problem; there are two variants. The second if formulated in H(div) and ellipticity of the leading term can only be established in the subspace of divergence-free functions.
• March 26: Error bounds for mixed finite element methods: Counter parts of Cea's lemma and the use of a Strang lemma is also required in some cases. H(div)-conforming spaces due to Raviart and Thomas and H(curl)-conforming spaces of Nedelec. what is relevant in these cases is continuity of the normal component and the tangential components, respectively. A discussion of how to establish inf-sup stability for pairs of finite element spaces; in some cases as for flow in porous media it can be straightforward while for incompressible Stokes equations, it is much harder. A few words on the Stokes case; to be continued.
• April 2: Proving inf-sup stability for Stokes equations following Rolf Stenberg. Local problems on macro-elements can be shown to be inf-sup stable and global bounds can then be derived from the local bounds. A short introduction to solvers for large linear algebraic systems of equations arising in finite element work. For sparse Cholesky upper and lower bound on the complexity of these problems are known: work grows in proportion to N^{3/2} for problems in 2D and in proportion to N^2 for 3D. Here N is the number of unknowns.
• April 9: