A posteriori error analysis of an augmented fully

XXV COMCA
Congreso de Matemática Capricornio
2,3,4 y 5 de Agosto de 2016, Antofagasta, Chile
A posteriori error analysis of an augmented
fully–mixed FEM for the Boussinesq problem
Eligio Colmenares∗
Centro de Investigación en Ingeniería Matemática (CI2 MA) and
Departamento de Ingeniería Matemática
Universidad de Concepción
Concepción, Chile
Abstract
In this talk we present an a posteriori error analysis for a high–order quasi–optimally convergent
augmented fully–mixed finite element method introduced and analyzed in an earlier work of us to
numerically simulate heat driven flows in the Boussinesq approximation setting. Our approach
there incorporates as additional unknowns a modified pseudostress tensor and an auxiliary vector in the governing fluid and heat equations, respectively, the pressure is then eliminated by its
own definition, and redundant Galerkin terms are included to the resulting weak formulation.
The corresponding solvability analysis, its discretization, and the convergence of the latter were
stated, and for any conforming family of finite element subspaces. Optimal order a priori error
estimates were particularly proven by using Raviart–Thomas elements for the aforementioned
auxiliary unknowns, and Lagrange elements for the velocity and the temperature. Here we
propose a reliable and efficient, fully local and computable, residual–based a posteriori error
estimator in two and three dimensions. Standard arguments based on duality techniques, stable
Helmholtz decompositions, and well–known results from previous a posteriori error analyses of
related mixed schemes are the main underlying tools used in our methodology. Numerical experiments validate the expected behavior of the associated adaptive algorithm and illustrate the
accuracy improvement of the technique for approximating not only the principal unknowns but
also several other physically relevant post-processed variables, such as the pressure, the vorticity
fluid, the shear–stress tensor, and the velocity and the temperature gradient.
Joint work with:
Gabriel N. Gatica1 , CI2 MA y Departmento de Ingeniería Matemática, Universidad de Concepción, Concepción, Chile.
Ricardo Oyarzúa2 , GIMNAP-Departamento de Matemática, Universidad del Bío-Bío y CI2 MA –
Universidad de Concepción, Concepción, Chile.
∗ Partially supported by the Becas–Chile programme for foreign students and Centro de Investigación en Ingeniería
Matemática (CI2 MA), Universidad de Concepción, e-mail: eligio@ci2ma.udec.cl
1 Partially supported by CONICYT-Chile through BASAL project CMM, Universidad de Chile, project Anillo
ACT1118 (ANANUM), and project Fondecyt 11121347, e-mail: ggatica@ci2ma.udec.cl
2 Partially supported by Universidad del Bío-Bío through DIUBB project 120808 GI/EF, e-mail:
royarzua@ubiobio.cl
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References
[1] M. Álvarez, G. N. Gatica, R. Ruíz–Baier, A posteriori error analysis for a viscous
flow–transport problem, ESSAIM: Mathematical Modelling and Numerical Analysis, DOI:
http://dx.doi.org/10.1051/m2an/2016007.
[2] E. Colmenares, G. N. Gatica, R. Oyarzúa, An augmented fully-mixed finite element
method for the stationary Boussinesq problem, Calcolo, to appear. DOI: 10.1007/s10092-0160182-3.
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Equations, vol. 32, 2, pp. 445-478, (2016).
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