# IJPAM: Volume 47, No. 4 (2008)

**ADAPTIVE CALCULATION OF VARIABLE COEFFICIENTS**

ELLIPTIC DIFFERENTIAL EQUATIONS VIA WAVELETS

ELLIPTIC DIFFERENTIAL EQUATIONS VIA WAVELETS

School of Mathematical Sciences

Faculty of Exact Sciences

Tel Aviv University

P.O. Box 39040, Ramat Aviv, Tel Aviv, 69978, ISRAEL

Faculty of Computer Science

Technion - Israel Institute of Technology

Haifa, 32000, ISRAEL

**Abstract.**We propose a solver for 1-D strictly elliptic linear PDE's with non-constant coefficients of the form
.
We combine a sparse multiplication algorithm with a diagonally preconditioned conjugate gradient (CG) method.
We use sparse data structures to take advantage of the complexity of the algorithm,
where is the number of significant coefficients (i.e. above a certain threshold) required for a
given accuracy.

We show that the usage of a sparse multiplication in wavelet space
rather than in the
original physical space can speed up the performance of the sparse solver by a factor of 20.
We present an algorithm and numerical results for an adaptive multiplication scheme that can rapidly
solve the equation above. We explore, in detail, how the accuracy of the
wavelet-based multiplication is affected by different input parameters for the algorithm. We
integrated a sparse multiplication into the PDE solver. The relation between the
performance of the solver and the parameters of the wavelet based sparse multiplication is also studied.
This integration allowed us to extend the fast adaptive algorithms to achieve numerical solutions of
linear non-constant coefficient differential equations.
One-dimensional numerical examples for using stand alone sparse
multiplication and for the differential equations solver are
presented. This work, which is based on [#!a1!#], extends it.

**Received: **August 20, 2008

**AMS Subject Classification: **35J15

**Key Words and Phrases: **solver for 1-D strictly elliptic linear PDE, wavelet space, sparse data structures, numerical results for an adaptive multiplication scheme

**Source:** International Journal of Pure and Applied Mathematics

**ISSN:** 1311-8080

**Year:** 2008

**Volume:** 47

**Issue:** 4