# Summary of the course

## An Introduction to Partition of Unity-Based

Finite Element Methods

### Carlos Armando Duarte

#### University of Illinois at Urbana-Champaign, USA

**Course Objectives**

Introduce and develop a thorough understanding of partition of unity approximations – generalized and extended finite element approximations: Strengths, advantages, a-priori error estimates, implementation issues and applications.

**Course Outline**

Partition of unity approximations

Open cover, partition of unity and reproducing condition

Finite element Partition of unity

Shepard partition of unity

Partition of unity shape functions

Reproducing condition

Pasting of local approximations

A-priori error estimates

Generalized/eXtended FEM shape functions in 1-, 2- and 3-dimensions

*h*and*p*extensionsCompleteness

Conditioning, linear dependence and solution of equations

Applications of GFEM

Weak and strong discontinuities

Singularities in 2- and 3-dimensions

Arbitrary cracks and crack propagation

Multiscale problems

Construction of scale-bridging enrichment functions