# Analysis of finite element methods for second order boundary value problems using mesh dependent norms

## Abstract

A new approach to the analysis of finite-element methods is presented which is based on C/sup 0/-finite elements for the approximate solution of second-order boundary-value problems in which error estimates are derived directly in terms of two mesh-dependent norms that are closely related to the L/sub 2/ norm and to the second-order Sobolev norm, respectively, and in which there is no assumption of quasi-uniformity on the mesh family. This is in contrast to the usual analysis, in which error estimates are first derived in the first-order Sobolev norm and subsequently are derived in the L/sub 2/ norm in the second-order Sobolev norm--the second-order Sobolev norm estimates being obtained under the assumption that the functions in the underlying approximating subspaces lie in the second-order Sobolev space and that the mesh family is quasi-unform.

- Authors:

- Publication Date:

- Research Org.:
- Wisconsin Univ., Madison (USA). Mathematics Research Center

- OSTI Identifier:
- 6456041

- Alternate Identifier(s):
- OSTI ID: 6456041

- Report Number(s):
- ORO-3443-78

- DOE Contract Number:
- EY-76-S-05-3443

- Resource Type:
- Technical Report

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; BOUNDARY-VALUE PROBLEMS; NUMERICAL SOLUTION; MECHANICS; CALCULATION METHODS; COMBUSTION; DIFFERENTIAL EQUATIONS; FINITE ELEMENT METHOD; FLUID MECHANICS; FRACTURES; CHEMICAL REACTIONS; EQUATIONS; FAILURES; OXIDATION; THERMOCHEMICAL PROCESSES 658000* -- Mathematical Physics-- (-1987)

### Citation Formats

```
Babuska, I., and Osborn, J.
```*Analysis of finite element methods for second order boundary value problems using mesh dependent norms*. United States: N. p., 1979.
Web.

```
Babuska, I., & Osborn, J.
```*Analysis of finite element methods for second order boundary value problems using mesh dependent norms*. United States.

```
Babuska, I., and Osborn, J. Thu .
"Analysis of finite element methods for second order boundary value problems using mesh dependent norms". United States.
```

```
@article{osti_6456041,
```

title = {Analysis of finite element methods for second order boundary value problems using mesh dependent norms},

author = {Babuska, I. and Osborn, J.},

abstractNote = {A new approach to the analysis of finite-element methods is presented which is based on C/sup 0/-finite elements for the approximate solution of second-order boundary-value problems in which error estimates are derived directly in terms of two mesh-dependent norms that are closely related to the L/sub 2/ norm and to the second-order Sobolev norm, respectively, and in which there is no assumption of quasi-uniformity on the mesh family. This is in contrast to the usual analysis, in which error estimates are first derived in the first-order Sobolev norm and subsequently are derived in the L/sub 2/ norm in the second-order Sobolev norm--the second-order Sobolev norm estimates being obtained under the assumption that the functions in the underlying approximating subspaces lie in the second-order Sobolev space and that the mesh family is quasi-unform.},

doi = {},

journal = {},

number = ,

volume = ,

place = {United States},

year = {1979},

month = {2}

}