  This paper develops and analyzes a general iterative framework for solving
parameter-dependent and random convection-diffusion problems. It is inspired by
the multi-modes method of [7,8] and the ensemble method of [20] and extends
those methods into a more general and unified framework. The main idea of the
framework is to reformulate the underlying problem into another problem with
parameter-independent convection and diffusion coefficients and a
parameter-dependent (and solution-dependent) right-hand side, a fixed-point
iteration is then employed to compute the solution of the reformulated problem.
The main benefit of the proposed approach is that an efficient direct solver
and a block Krylov subspace iterative solver can be used at each iteration,
allowing to reuse the $LU$ matrix factorization or to do an efficient
matrix-matrix multiplication for all parameters, which in turn results in
significant computation saving. Convergence and rates of convergence are
established for the iterative method both at the variational continuous level
and at the finite element discrete level under some structure conditions.
Several strategies for establishing reformulations of parameter-dependent and
random diffusion and convection-diffusion problems are proposed and their
computational complexity is analyzed. Several 1-D and 2-D numerical experiments
are also provided to demonstrate the efficiency of the proposed iterative
method and to validate the theoretical convergence results.
