### 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.

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.

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**Asociación Argentina de Mecánica Computacional**Güemes 3450

S3000GLN Santa Fe, Argentina

Phone: 54-342-4511594 / 4511595 Int. 1006

Fax: 54-342-4511169

E-mail: amca(at)santafe-conicet.gov.ar

**ISSN 2591-3522**