Department of Mathematics, UCI

Partial differential equations were discretized using the mortar finite element method, where the mortar space contains piecewise quadratic and cubic functions. We first proved the wellposedness of the saddle point system. To solve this saddle point system, the existing domain decomposition (DD) algorithm of the linear system solver requires the communication of both the interface solution and the interface residual of every local problem. We have developed a new algorithm where only the interface solution is communicated, to accommodate globally nonconforming meshes. The resulting communication complexity is reduced. The scalability and parallel efficiency of this new algorithm were tested with a highly adaptive mesh. As the number of processors increases, linear and logarithmic speed-up of the solving time were observed with this new linear system solver, for the convection-diffusion equation and the Poisson equation respectively.

Three-dimensional dynamic simulation in computational biology provides an emerging field for the application of efficient distributed linear system solvers. We developed a 3D continuum model to investigate the role of structural and functional cellular components in regulating synchronized calcium signaling (SCS), characterized by high gradient near the t-tubule membrane and low gradient in the cytoplasm along the transverse direction, which enables ventricular myocyte to respond rapidly and forcefully to electrical andchemical stimuli. The distributed linear system solver improved the simulation speed by ~10 folds. Simulation results suggest that both t-tubule structure and the spatially heterogeneous distribution of calcium-handling-proteins are important for SCS. The model also predicts that two aspects of heterogeneous distribution are required: the concentration of calcium-handling-proteins in the t-tubule membrane to be ~6 times of that in the surface membrane; and the concentration of L-type calcium channels, in the cytoplasmic end of the t-tubule, to be ~2.3 times of that in the surface membrane end. These results have provided a foundation for further studies on the effects of three- dimensional t-tubule geometry and ion channel distribution on calcium dynamics.