TAM 574: Advanced Finite Element Methods

Class Description:

Advanced theory and applications of the finite-element method, as needed for research in computational science and engineering: applications to mechanics of solids and fluids, thermal problems, etc.; variational foundations of the finite-element method, error estimates, and adaptive analysis; finite-element methods for parabolic and hyperbolic problems; mixed finite-element methods; and applications to systems of equations. Same as CSE 517. Prerequisite: One of TAM 470, CEE 570, CS 455, ME 471, or an equivalent. 4 graduate hours.

Textbook:

Recommended: Brenner and Scott, The Mathematical Theory of the Finite Method, New York: Springer-Verlag (1994).
Oden and Demkowicz, Applied Functional Analysis, CRC Press.
Hughes, The Finite Element Method: Linear Static and Dynamic Finite Element Analysis, Dover.

Topics:

Survey of topics in real and functional analysis (12 hr)
Set theory; Lebesgue integration; vector spaces, inequality theorems; continuity, boundedness, bounded inverse theorem, convergence, completeness; Hilbert spaces, weak derivatives, Sobolev spaces; duality, Riesz representation theorem

Elliptic problems: variational forms, abstract formulation (12 hr)
Weak forms of elliptic boundary-value problems; energy norm; Dirichlet, Neumann, and mixed boundary conditions; finite-element discretization; abstract formulation; error equation; Lax-Milgram theory; applications to elasticity, incompressible flow

Approximation theory; error estimates, regularity (4 hr)
Interpolation error, a-priori error estimates, regularity of solutions

Post-processing and adaptive analysis (6 hr)
Recovery of secondary variables; a-posteriori error indicators and estimates; h-, p- and hp-adaptive analysis methods, hierarchic shape functions

Parabolic problems (6 hr)
Continuous and semi-discrete variational formulations; stability; error estimates, integration rules; mass lumping, modal analysis; time-discontinuous Galerkin method; applications to transient conduction problems

Hyperbolic problems (7 hr)
Critique of Galerkin, artificial diffusion, and upwinding methods; consistent approximations: streamline upwind Petrov|ndash;Galerkin, Galerkin least squares, and discontinuous Galerkin methods; applications to fracture and advection-diffusion problems

Mixed finite elements; incompressible and nearly incompressible media (8 hr)
Mixed variational principles: elasticity, Stokes problem; abstract continuous problem; discrete problem; existence and uniqueness; inf-sup (Babuska-Brezzi) condition; examples of mixed elements; penalty, reduced-integration, and augmented Lagrangian methods

Additional applications and theoretical topics (3 hr)

Midterm Exam (2 hr)

TOTAL HOURS: 60

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TAM