Name: RAMONI ZANCANELA SEDANO AZEVEDO

Publication date: 29/09/2023
Advisor:

Name	Role
ISAAC PINHEIRO DOS SANTOS	Advisor

Examining board:

Name	Role
ISAAC PINHEIRO DOS SANTOS	Advisor

Summary: In this work we present a numerical study of the Dynamic Diffusion (DD) method for solving diffusion-
convection-reaction equations with dominant convection and reaction problems. We discuss the variational
formulation of the method for transient and stationary problems in two- and three-dimensional domains. The DD
method is a multiscale model where standard finite element spaces are enriched with bubble functions to add
stability properties to the numerical model and which incorporates into the multiscale formulation a nonlinear
dissipative operator acting isotropically on both scales of the discretization. We present three new ways to obtain
the artificial diffusion present in the diffusive operator, named DD2, DD3 and DD4, since we call DD1 the default
method. The DD1 method for two-dimensional problems results in good solutions compared to other known
stabilized methods. This motivated us to apply the method to three-dimensional problems. We observed that the
choices of the characteristic mesh length have a great influence on the quality of the approximate solution, being
necessary to choose according to the problem that is approached. We consider only the stationary cases, version
DD1 and DD2 of the method, we carry out a convergence study using the norms L2 () e H1 () of Sobolev
spaces to estimate the order of convergence. The study was carried out using two- and three-dimensional dominant
convection problems, considering different set of meshes to evaluate whether the convergence rate of the DD1 and
DD2 methods suffer any impact due to the different characteristics of the meshes. Numerical results show optimal
convergence rates in all cases considered. Furthermore, we perform numerical experiments using the DD1 and DD2
versions for the stationary problems considering two- and three-dimensional problems. As for the transient
problems, we carried out experiments with the four versions of the method, DD1, DD2, DD3, DD4, also for two- and
three-dimensional problems. Numerical results are compared with those obtained using the Consistent Approximate
Upwind (CAU) method. The solutions obtained by the DD2 method are less diffusive than the CAU and DD1 method
solutions for stationary problems and the solution presented by the DD4 method presents greater accuracy among
the transient problems. The discrete problem is stored in the known Compressed Sparse Row (CSR) data structures,
the non-linear systems are solved by Picard fixed point iteration and the linear systems are solved by the
preconditioned GMRES method.

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