We consider the solution of the system of linear algebraic equations which arises from the finite element discretization of boundary value problems associated to the differential operator I−graddiv. The natural setting for such problems is in the Hilbert space H (div) and the variational formulation is based on the inner product in H (div). We show how to construct preconditioners for these equ...

Single-stage or single-step high-order temporal discretizations of partial differential equations (PDEs) have shown great promise in delivering accuracy time with efficient use computational resources. There has been much success developing such methods for finite volume method (FVM) PDEs. The Picard Integral formulation (PIF) recently made single-stage accessible difference (FDM) discretizatio...

We conduct experimental study on numerical solution of the two dimensional convection-diiusion equation discretized by three nite diierence schemes: the traditional central diier-ence scheme, the standard upwind scheme and the fourth-order compact scheme. We study the computed accuracy achievable by each scheme, the algebraic properties of the coeecient matrices arising from diierent schemes an...

We examine nonstationary inverse problems in which the time evolution of the unknown quantity is modelled by a stochastic partial differential equation. We consider the problem as a state estimation problem. The time discrete state evolution equation is exact since the solution is given by an analytic semigroup. For the practical reasons the space discretization of the time discrete state estim...

In diffuse optical tomography (DOT), the discretization error in the numerical solutions of the forward and inverse problems results in error in the reconstructed optical images. In this first part of our work, we analyse the error in the reconstructed optical absorption images, resulting from the discretization of the forward and inverse problems. Our analysis identifies several factors which ...

ACKNOWLEDGEMENTS I would like to truly thank all of my committee members for their encouragement and guidance in the preparation of this thesis. I especially want to express my gratitude to Dr. Kevin Clarno for his insight and assistance. My advisor, Dr. Cassiano de Oliveira, has been immensely helpful with his continuous encouragement and unique perspectives. Perhaps most of all, I want to tha...

We present a new three-dimensional steady-state saturated groundwater-flow forward-simulator with full conductivity tensors using a nineteen-points block-centered finite-difference method. Hydraulic conductivity tensors are defined at the block interfaces eliminating the need to average conductivity tensors at adjacent blocks to approximate their values at the interfaces. The capabilities of th...

We use the idea of nonstandard finite difference methods to derive the discrete variational integrators for multisymplectic PDEs. We obtain a nonstandard finite difference variational integrator for linear wave equation with a triangle discretization and two nonstandard finite difference variational integrators for the nonlinear Klein-Gordon equation with a triangle discretization and a square ...