# Semi-orthogonal wavelets for elliptic variational problems

## Abstract

In this paper the authors give a construction of wavelets which are (a) semi-orthogonal with respect to an arbitrary elliptic bilinear form a({center_dot},{center_dot}) on the Sobolev space H{sub 0}{sup 1}((0, L)) and (b) continuous and piecewise linear on an arbitrary partition of [0, L]. They illustrate this construction using a model problem. They also construct alpha-orthogonal Battle-Lemarie type wavelets which fully diagonalize the Galerkin discretized matrix for the model problem with domain IR. Finally they describe a hybrid basis consisting of a combination of elements from the semi-orthogonal wavelet basis and the hierarchical Schauder basis. Numerical experiments indicate that this basis leads to robust scalable Galerkin discretizations of the model problem which remain well-conditioned independent of {epsilon}, L, and the refinement level K.

- Authors:

- Vanderbilt Univ., Nashville, TN (United States). Mathematics Dept.
- Sandia National Labs., Albuquerque, NM (United States)

- Publication Date:

- Research Org.:
- Sandia National Labs., Albuquerque, NM (United States)

- Sponsoring Org.:
- USDOE Office of Financial Management and Controller, Washington, DC (United States); National Science Foundation, Washington, DC (United States)

- OSTI Identifier:
- 672012

- Report Number(s):
- SAND-98-0975C; CONF-980434-

ON: DE98004760; BR: YB0100000; TRN: AHC2DT07%%171

- DOE Contract Number:
- AC04-94AL85000

- Resource Type:
- Conference

- Resource Relation:
- Conference: International wavelet conference Tangier 98, Tangier (Morocco), 13 Apr 1998; Other Information: PBD: Apr 1998

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 99 MATHEMATICS, COMPUTERS, INFORMATION SCIENCE, MANAGEMENT, LAW, MISCELLANEOUS; DIFFERENTIAL EQUATIONS; VARIATIONAL METHODS; GALERKIN-PETROV METHOD; MATRICES; MATHEMATICAL SPACE; TRANSFORMATIONS; ALGORITHMS

### Citation Formats

```
Hardin, D.P., and Roach, D.W.
```*Semi-orthogonal wavelets for elliptic variational problems*. United States: N. p., 1998.
Web.

```
Hardin, D.P., & Roach, D.W.
```*Semi-orthogonal wavelets for elliptic variational problems*. United States.

```
Hardin, D.P., and Roach, D.W. Wed .
"Semi-orthogonal wavelets for elliptic variational problems". United States. https://www.osti.gov/servlets/purl/672012.
```

```
@article{osti_672012,
```

title = {Semi-orthogonal wavelets for elliptic variational problems},

author = {Hardin, D.P. and Roach, D.W.},

abstractNote = {In this paper the authors give a construction of wavelets which are (a) semi-orthogonal with respect to an arbitrary elliptic bilinear form a({center_dot},{center_dot}) on the Sobolev space H{sub 0}{sup 1}((0, L)) and (b) continuous and piecewise linear on an arbitrary partition of [0, L]. They illustrate this construction using a model problem. They also construct alpha-orthogonal Battle-Lemarie type wavelets which fully diagonalize the Galerkin discretized matrix for the model problem with domain IR. Finally they describe a hybrid basis consisting of a combination of elements from the semi-orthogonal wavelet basis and the hierarchical Schauder basis. Numerical experiments indicate that this basis leads to robust scalable Galerkin discretizations of the model problem which remain well-conditioned independent of {epsilon}, L, and the refinement level K.},

doi = {},

journal = {},

number = ,

volume = ,

place = {United States},

year = {1998},

month = {4}

}