DOMAIN DECOMPOSITION AND THE NUMERICAL SOLUTION OF SYSTEMS OF DIFFERENTIAL EQUATIONS WITH THE TAU METHOD

Youssef M. SAMROUT

The work discussed here is on the numerical application of the Tau Method to problems in ordinary and partial differential equations which may include singularities and a nontrivial domain.

This work is mainly based on the study and application of two recently sug­gested formulations of the Tau Method, the so-called, "The Operational" and "The Tau-Lines" approaches.

In the former case, applications have been carried out on a wide range of ini­tial, boundary and mixed boundary value problems, defined by linear and non linear systems of ordinary differential equations. Very accurate approximate solutions have been obtained.

In the case of extended intervals, -segmentation of the range of the indepen­dent variable plays an important role in obtaining an approximate solution of high accuracy. This approach was followed developing suitable software and applying it to a variety of problems.

Also in the case of segmentation, This work has discussed from a computational view­point, the dependence of the Tau Method approximation error on the length of the subintervals. It has been found that such dependence is of exponential type. This result suggested analytical investigations which have confirmed the conjecture.

The bivariate formulation of the operational approach and of the Tau-Lines methods have been applied to linear initial and boundary value problems in partial differential equations.

Similarly, in the case of partial differential equations defined over domains with a complex geometry, domain decomposition techniques, introduced by Ortiz and Pun in the form of a Tau-elements approach, have been successfully im­plemented and applied to the Tau Method. Some domains have been considered of non-rectangular shape and obtained encouraging results. The study has also discussed domains limited by simple curves and applied these tech­niques to solve a singular partial differential equation related to cracks in which the crack line is not rectilinear.

This research also developed a technique for the automatic generation of matching conditions for subdomains in the case of domain decomposition. This tech­nique is of interest when applied to T-shaped and L-shaped domains and to domains limited by a curved boundary.