# Advanced Numerical Methods in the Mathematical Sciences

## Randolph Bank, University of California at San Diego Some Recent Results for Adaptive Finite Elements

### Abstract

We will discuss our ongoing investigation of adaptive strategies for finite element equations. We show first that under modest assumptions, any robust and efficient a posteriori error indicator must behave as a simple interpolation error for the exact finite element solution. We then use the interpolation error to study several popular $$h$$ and $$hp$$ adaptive algorithms.