Bulletin of the
Korean Mathematical Society BKMS

We introduce and analyze a naturally parallelizable fre-quency--domain method for parabolic problems with a Neumann boundary condition. After taking the Fourier transformation of given equations in the space--time domain into the space--frequency domain, we solve an indefinite, complex elliptic problem for each frequency. Fourier inversion will then recover the solution of the original problem in the space-time domain. Existence and uniqueness of a solution of the transformed problem corresponding to each frequency is established. Fourier invertibility of the solution in the frequency--domain is also examined. Error estimates for a finite element approximation to solutions of transformed problems and full error estimates for solving the given problem using a discrete Fourier inverse transform are given.

Keywords: parabolic problems, Neumann boundary conditions, frequency--domain methods, finite element methods, parallel algorithm, Fourier transform

MSC numbers: Primary 65N30, Secondary 35K20