Center for Mathematics, Fundamental Applications and Operations Research

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Continuous/discontinuous Galerkin approximations for a fourth-order nonlinear problem

Publication . Lopes, N.D.; de Oliveira, H. B.

We introduce a Continuous/Discontinuous Galerkin Finite Element Method (CDFEM) with interior penalty terms to solve a nonlinear fourth-order problem that appears in the analysis of the confinement of fluid flows governed by the Stokes system. For the associated continuous and discrete problems, we prove the existence and uniqueness of weak solutions. Consistency, stability and convergence of the method are shown analytically. To show the applicability and robustness of the numerical model, several test cases are performed.

2021Journal article

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Kelvin-Voigt equations for incompressible and nonhomogeneous fluids with anisotropic viscosity, relaxation and damping

Publication . Antontsev, S. N.; de Oliveira, H. B.; Khompysh, Kh.

In this work, we consider the nonlinear initial-boundary value problem posed by the Kelvin-Voigt equations for non-homogeneous and incompressible fluid flows with fully anisotropic diffusion, relaxation and damping. Moreover, we assume that the momentum equation is perturbed by a damping term which, depending on whether its signal is positive or negative, may account for the presence of a source or a sink within the system. In the particular case of considering this problem with a linear and isotropic relaxation term, we prove the existence of global and local weak solutions for the associated initial-boundary value problem supplemented with no-slip boundary conditions. When the damping term describes a sink, we establish the conditions for the polynomial time decay or for the exponential time decay of these solutions.

2022Journal article

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