Accelerated Dirichlet-Robin Alternating Algorithm for Solving the Cauchy Problem for an Elliptic Equation using Krylov Subspaces

Abstract

In this thesis, we study the Cauchy problem for an elliptic equation. We use Dirichlet-Robin iterations for solving the Cauchy problem. This allows us to include in our consideration elliptic equations with variable coefficient as well as Helmholtz type equations. The algorithm consists of solving mixed boundary value problems, which include the Dirichlet and Robin boundary conditions. Convergence is achieved by choice of parameters in the Robin conditions. i To my wonderful and loving parents, Mr. and Mrs. David Mutai and my siblings, your compassion, strength, and unconditional love are what carry me throughout my research. Thank you for being there by my side when times are tough. To my lovely son, Asier Kiptoo thank you very much for your patiences and endurances during my studies. I also take this opportunity to thank the entire family of Sing'oei for taking good care of my son.Finally, my most sincere appreciation go to ISP and the Eastern African Universities Mathematics Programme (EAUMP) for the financial support. I offer my sincere thanks to my coordinators; Lief Abrahamson, Patrick G. Weke, Jared Ongaro and the entire team working at ISP who have tirelessly ensure my studies and stay in Sweden is comfortable.

Regularization methods for solving Cauchy problems for elliptic and degenerate elliptic equations

“…Picking the appropriate α requires a formal parameter choice rule. One such rule is the Discrepancy principle, see [13] and also see [9], which can be proven to give a convergent regularization scheme, i.e. ∥x δ α(δ) − x∥ → 0 as δ → 0.…”

“…where ξ = g 0 +µ 0 f 0 −(−∂ y ũ(x, 0)+µ 0 ũ(x, 0))), see [9]. We also evaluate the adjoint operator T * of the operator T with respect to the inner products in H −1/2 (Γ 0 ) and H −1/2 (Γ 1 ).…”