Introduction to Finite Element Methods

3-0-9 units · Graduate

Ordinary differential equation boundary value problems: 2nd-order, 4th-order spatial operators; eigenproblems. Partial differential equations for scalar fields: elliptic, parabolic, hyperbolic. Strong statement, weak form, minimization principle. Rayleigh-Ritz, Galerkin projection. Numerical interpolation, integration, differentiation; best-fit. Finite element method for spatial discretization in one and two space dimensions: formulation (linear, quadratic approximation), mesh generation, bases and discrete equations, uniform and adaptive refinement, a priori and a posteriori error estimates, sparse solvers, implementation, testing. Finite difference-finite element methods for mixed initial-boundary value problems; nonlinear problems and Newton iteration; linear elasticity. Applications in heat transfer and structural analysis. Assignments require MATLAB coding. Students taking graduate version complete additional work.