Least-Squares Formulations Applied To Parabolic Equations

Abstract

During the last decade finite element least-squares formulation has been widely used
for solving differential equations. Applied to stationary Poisson problem, written as a
first order system, it gives H1 norm convergence for the scalar field and Hdiv for fluxes
without satisfying the compatibility requirements between the spaces used to approximate
these variables. Adding non rotational condition we get H1 convergence for both fields
involved in the problem.
In this work we present least-squares semi-discrete formulations applied to the transient
heat equations written in temperature and flux. Three time weight were used in the
functional definitions resulting in a totally implicit formulation, a weighted formulation
where the evolution equation was weighted by a θ factor between t and t+Δt time step
and the constitutive equation relating the scalar quantity and its fluxp osed in time t+Δt
and a third one named here as θ-least-squares formulation (θEFMQ).
The three here proposed formulations are applied to an example and convergence errors
curves are shown and discussed.