Parallel numerical algorithms for the solution of diffusion problems

The purpose of this thesis is to determine the most effective parallel algorithm for the solution of the parabolic differential equations characteristic of diffusion problems. The primary aim is to apply the chosen algorithm to obtain solutions to the equations governing the operation of membrane-covered oxygen sensors, known as Clark electrodes, which are used for monitoring the oxygen concentration of blood. The boundary conditions of this problem require the development of a singularity correction technique. A brief history of electrochemical sensors leading to the development of the Clark electrode is given, together with the two-dimensional equations and boundary conditions governing its operation. A locally valid series expansion is derived to take care of the boundary singularity, together with a robust method of matching this to the finite difference approximation. Parallel implementations of three representative numerical algorithms applied to a simple model problem are compared by extending Leland's parallel effectiveness model. The chosen parallel algorithm is combined with the singularity correction to obtain a solution to the Clark electrode problem. Numerical experiments show this solution to achieve the required accuracy. Previous one-dimensional models of the Clark electrode are shown to be inadequate before the two-dimensional model is used to examine the variation of operation with design. The understanding gained allows us to demonstrate the advantages of pulse amperometry over steady-state techniques, and to suggest the most appropriate method and design for use in in vivo clinical monitoring.