Advanced Finite Element Methods

The outline for the course is:

1. Overview of the standard finite element method (FEM)

2. The problem of linear elastostatics in one dimension

• The strong (classical) form

• The weak (variational) form

• The Galerkin formulation

• Orthogonality of Galerkin error

• The finite element discrete equations

• Iso-parametric mapping

3. The elastostatics problem in higher dimensions

• Strong form

• Weak form

• Discrete formulation

4. Heat conduction

5. The p-version of the finite element method (p-FEM)

• p-hierarchical elements in 1-D

• p-FEM in higher dimensions

• Integration and geometrical mapping

• Prescribing essential (Dirichlet) boundary conditions

6. h, p, and hp convergence

• Problem classification based on regularity

• A priori error estimates

• A posteriori error estimate for h-FEM

7. Computational aspects

• Finite element implementation

• The assembly process and numerical integration

• Checking the implementation (high-order patch tests and construction of manufactured solutions)