##### Department of Mathematics,

University of California San Diego

****************************

### Special Recruitment Colloquium

## Long Chen

#### Penn State University

## Optimal interpolation error estimates and applications

##### Abstract:

In this talk, we first present an interpolation error estimate in $L^p$ norm ($1\leq p\leq \infty$) for finite element simplicial meshes in any spatial dimensions and then discuss its applications to computational geometry and numerical solution of PDEs. We show that an asymptotically optimal error estimate can be obtained under near optimal meshes. A sufficient condition for a mesh to be nearly optimal is that it is quasi-uniform under a new metric defined by a modified Hessian matrix of the function to be interpolated. We further show such estimates are in fact asymptotically sharp for strictly convex functions. \vskip .1in \noindent To illustrate the useful of our results, we present an efficient polygonal curve simplification algorithm which improve the computational cost to be optimal. We also briefly discuss some interesting and related problems in the computational geometry, such as sphere covering and optimal polytope approximation of convex bodies. \vskip .1in \noindent The above interpolation error estimate is useful for approximating functions with anisotropic singularity. Thus it can be applied to convection diffusion problem with small diffusion parameter $\varepsilon$, of which solutions usually present boundary layers or interior layers. For a type of 1-D problems, we have carefully designed a special streamline diffusion finite element method whose discretization error is shown to be uniformly governed by the interpolation error in maximum norm. For problems in multidimensions, we shall discuss some practical issues in the algorithms especially the homotopy with respect to the parameter $\varepsilon$.

Host: Randy Bank

### January 6, 2005

### 1:00 PM

### AP&M 6438

****************************