CATALOG DATA:
Fundamental aspects of the finite element method are developed and applied to the solution of PDE's encountered in science and engineering. Solution strategies for parabolic, elliptic and hyperbolic equations are explored.

TEXTBOOKS:
Celia, M.A. and Gray, W.G., Numerical Methods for Differential Equations, Prentice Hall, Englewood Cliffs, 1991.

Topics:

• Finite difference methods: Derivation of 1-D FD formulae; variable coefficients; unequal spacing; extrapolation; derivative boundary conditions; 2-D FD formulae.
• Solution to the transient 1D diffusion equation: Explicit methods; implicit methods.
• Accuracy and stability analysis: Truncation error analysis; heuristic stability analysis, stability analysis by von Neumann's method.
• Solution of the transient 2D diffusion equation: Explicit methods, implicit methods, ADI method, irregular boundaries.
• Solution to the convection-diffusion equation using FD methods: Standard solutions, control of spurious modes, upwinded solutions, higher order upwinding, truncation error and Fourier stability analysis, evaluation of accuracy by Fourier analysis.
• Method of weighted residuals: Admissibility, norms, orthogonality and completeness, collocation, least squares, Galerkin, subdomain, method of moments, least squares collocation.
• Symmetry and other properties of matrices and operators: Symmetry and self adjointness, positive definite matrices and operators, derivation of boundary conditions associated with an operator.
• Weak formulations: Sobelov space, fundamental weak form, symmetrical weak form, use of localized functions, weak forms in 2D, time dependent problems.
• The Finite Element Method: General steps, derivation of 1D C Lagrange elements, derivation of 1D C' Hermite elements, applications using cardinal and localized basis functions, numerical quadrature, time discretization, lumping.
• Solution of the convection-diffusion equation using FEM: Standard Galerkin, control of spurious oscillation, Petrov-Galerkin solutions, higher order Petrov-Galerkin solutions, truncation error analysis and Fourier analysis.
• FE method for 2D problems: derivation of interpolating basis for quadrilateral elements, derivation of interpolating basis for triangular elements, example applications.
• Improved accuracy through temporal and spatial refinement: Uniform refinement, selective refinement, mesh optimization, dynamic gridding.

ABET category content as estimated by faculty member who prepared the course description:
Engineering Science: 3 credits or 100%
Engineering Design: 0 credits

Prepared by: Professor J.J. Westerink
Last Update: March 27, 1994

Direct comments, questions, and corrections to amedept@nd.edu