Javier de Frutos

Javier de Frutos, Bosco García-Archilla, Volker John et al.

This paper studies inf-sup stable finite element discretizations of the evolutionary Navier--Stokes equations with a grad-div type stabilization. The analysis covers both the case in which the solution is assumed to be smooth and consequently has to satisfy nonlocal compatibility conditions as well as the practically relevant situation in which the nonlocal compatibility conditions are not satisfied. The constants in the error bounds obtained do not depend on negative powers of the viscosity. Taking into account the loss of regularity suffered by the solution of the Navier--Stokes equations at the initial time in the absence of nonlocal compatibility conditions of the data, error bounds of order $\mathcal O(h^2)$ in space are proved. The analysis is optimal for quadratic/linear inf-sup stable pairs of finite elements. We also consider the analysis of the fully discrete case with the backward Euler method as time integrator.

Javier de Frutos, Bosco García-Archilla, Volker John et al.

This paper studies non inf-sup stable finite element approximations to the evolutionary Navier--Stokes equations. Several local projection stabilization (LPS) methods corresponding to different stabilization terms are analyzed, thereby separately studying the effects of the different stabilization terms. Error estimates are derived in which the constants in the error bounds are independent of inverse powers of the viscosity. For one of the methods, using velocity and pressure finite elements of degree $l$, it will be proved that the velocity error in $L^\infty(0,T;L^2(Ω))$ decays with rate $l+1/2$ in the case that $ν\le h$, with $ν$ being the dimensionless viscosity and $h$ the mesh width. In the analysis of another method, it was observed that the convective term can be bounded in an optimal way with the LPS stabilization of the pressure gradient. Numerical studies confirm the analytical results.

Javier de Frutos, Víctor Gatón, Julia Novo

In this paper we study a discrete-time semidiscretization and a fully discretization (discrete-time, discrete-state) of an infinite time horizon noncooperative $N$-player differential game. We prove that as either the discretization time step or both time step and mesh size parameters approach zero the discrete value function approximates the value function of the differential game. Furthermore, the discrete Nash equilibrium is an $ε$-Nash equilibrium for the continuous-time differential game both in the discrete-time and fully discrete cases.

Javier de Frutos, Bosco García-Archilla, Julia Novo

A two-grid scheme based on mixed finite-element approximations to the incompressible Navier-Stokes equations is introduced and analyzed. In the first level the standard mixed finite-element approximation over a coarse mesh is computed. In the second level the approximation is postprocessed by solving a discrete Oseen-type problem on a finer mesh. The two-level method is optimal in the sense that, when a suitable value of the coarse mesh diameter is chosen, it has the rate of convergence of the standard mixed finite-element method over the fine mesh. Alternatively, it can be seen as a postprocessed method in which the rate of convergence is increased by one unit with respect to the coarse mesh. The analysis takes into account the loss of regularity at initial time of the solution of the Navier-Stokes equations in absence of nonlocal compatibility conditions. Some numerical experiments are shown.

Javier de Frutos, Bosco García-Archilla, Julia Novo

We consider two-grid mixed-finite element schemes for the spatial discretization of the incompressible Navier-Stokes equations. A standard mixed-finite element method is applied over the coarse grid to approximate the nonlinear Navier-Stokes equations while a linear evolutionary problem is solved over the fine grid. The previously computed Galerkin approximation to the velocity is used to linearize the convective term. For the analysis we take into account the lack of regularity of the solutions of the Navier-Stokes equations at the initial time in the absence of nonlocal compatibility conditions of the data. Optimal error bounds are obtained.

Javier de Frutos, Bosco García-Archilla, Julia Novo

A posteriori estimates for mixed finite element discretizations of the Navier-Stokes equations are derived. We show that the task of estimating the error in the evolutionary Navier-Stokes equations can be reduced to the estimation of the error in a steady Stokes problem. As a consequence, any available procedure to estimate the error in a Stokes problem can be used to estimate the error in the nonlinear evolutionary problem. A practical procedure to estimate the error based on the so-called postprocessed approximation is also considered. Both the semidiscrete (in space) and the fully discrete cases are analyzed. Some numerical experiments are provided.

Javier de Frutos, Bosco García-Archilla, Julia Novo

We obtain error bounds for a modified Chorin-Teman (Euler non-incremental) method for non inf-sup stable mixed finite elements applied to the evolutionary Navier-Stokes equations. The analysis of the classical Euler non-incremental method is obtained as a particular case. We prove that the modified Euler non-incremental scheme has an inherent stabilization that allows the use of non inf-sup stable mixed finite elements without any kind of extra added stabilization. We show that it is also true in the case of the classical Chorin-Temam method. The relation of the methods with the so called pressure stabilized Petrov Galerkin method (PSPG) is established. We do not assume non-local compatibility conditions for the solution.

Javier de Frutos, Bosco García-Archilla, Julia Novo

A modified Chorin-Teman (Euler non-incremental) projection method and a modified Euler incremental projection method for non inf-sup stable mixed finite elements are analyzed. The analysis of the classical Euler non-incremental and Euler incremental methods are obtained as a particular case. We first prove that the modified Euler non-incremental scheme has an inherent stabilization that allows the use of non inf-sup stable mixed finite elements without any kind of extra added stabilization. We show that it is also true in the case of the classical Chorin-Temam method. For the second scheme, we study a stabilization that allows the use of equal-order pairs of finite elements. The relation of the methods with the so called pressure stabilized Petrov Galerkin method (PSPG) is established. The influence of the chosen initial approximations in the computed approximations to the pressure is analyzed. Numerical tests confirm the theoretical results.

Javier de Frutos, Victor Gaton

This paper concerns the numerical solution of the finite-horizon Optimal Investment problem with transaction costs under Potential Utility. The problem is initially posed in terms of an evolutive HJB equation with gradient constraints. In Finite-Horizon Optimal Investment with Transaction Costs: A Parabolic Double Obstacle Problem, Day-Yi, the problem is reformulated as a non-linear parabolic double obstacle problem posed in one spatial variable and defined in an unbounded domain where several explicit properties and formulas are obtained. The restatement of the problem in polar coordinates allows to pose the problem in one spatial variable in a finite domain, avoiding some of the technical difficulties of the numerical solution of the previous statement of the problem. If high precision is required, the spectral numerical method proposed becomes more efficient than simpler methods as finite differences for example.