arXiv subjects

Gabriel Barrenechea

Publications and source records attributed to Gabriel Barrenechea.

5 recordsLinked to original sources

arXiv2026

A stabilized finite element method for a flow problem arising from 4D flow magnetic resonance imaging

In this work we propose, {analyze}, and validate a stabilized finite element method for a flow problem arising from the assessment of {4D Flow Magnetic Resonance Imaging quality}. Starting from the Navier-Stokes equation and splitting its velocity as the MRI-observed one (considered a datum) plus an ``observation error'', a modified Navier-Stokes problem is derived. This procedure allows us to estimate the quality of the measured velocity fields, while also providing an alternative approach to pressure reconstruction, thereby avoiding invasive procedures. Since equal-order approximations have become a popular choice for problems linked to pressure recovery from MRI images, we design a stabilized finite element method allowing equal-order interpolations for velocity and pressure. In the linearized version of the resulting model, we prove stability and (optimal order) error estimates and test the method with a variety of numerical experiments testing both the linearized case and the more realistic nonlinear one.

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arXiv2025

Finite element approximation to linear, second order, parabolic problems with $L^1$ data

We consider the approximation to the solution of the initial boundary value problem for the heat equation with right hand side and initial condition that merely belong to $L^1$. Due to the low integrability of the data, to guarantee well-posedness, we must understand solutions in the renormalized sense. We prove that, under an inverse CFL condition, the solution of the standard implicit Euler scheme with mass lumping converges, in $L^\infty(0,T;L^1(Ω))$ and $L^q(0,T;W^{1,q}_0(Ω))$ ($q<\tfrac{d+2}{d+1}$), to the renormalized solution of the problem.

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arXiv2024

Unconditionally stable, linearised IMEX schemes for incompressible flows with variable density

For the incompressible Navier--Stokes system with variable density and viscosity, we propose and analyse an IMEX framework treating the convective and diffusive terms semi-implicitly. This extends to variable density and second order in time some methods previously analysed for variable viscosity and constant density. We present three new schemes, both monolithic and fractional-step. All of them share the methodological novelty that the viscous term is treated in an implicit-explicit (IMEX) fashion, which allows decoupling the velocity components. Unconditional temporal stability is proved for all three variants. Furthermore, the system to solve at each time step is linear, thus avoiding the costly solution of nonlinear problems even if the viscosity follows a non-Newtonian rheological law. Our presentation is restricted to the semi-discrete case, only considering the time discretisation. In this way, the results herein can be applied to any spatial discretisation. We validate our theory through numerical experiments considering finite element methods in space. The tests range from simple manufactured solutions to complex two-phase viscoplastic flows.

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arXiv2023

A Nodally Bound-Preserving Finite Element Method

This work proposes a nonlinear finite element method whose nodal values preserve bounds known for the exact solution. The discrete problem involves a nonlinear projection operator mapping arbitrary nodal values into bound-preserving ones and seeks the numerical solution in the range of this projection. As the projection is not injective, a stabilisation based upon the complementary projection is added in order to restore well-posedness. Within the framework of elliptic problems, the discrete problem may be viewed as a reformulation of a discrete obstacle problem, incorporating the inequality constraints through Lipschitz projections. The derivation of the proposed method is exemplified for linear and nonlinear reaction-diffusion problems. Near-best approximation results in suitable norms are established. In particular, we prove that, in the linear case, the numerical solution is the best approximation in the energy norm among all nodally bound-preserving finite element functions. A series of numerical experiments for such problems showcase the good behaviour of the proposed bound-preserving finite element method.

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arXiv2019

Well-posedness and H(div)-conforming finite element approximation of a linearised model for inviscid incompressible flow

We consider a linearised model of incompressible inviscid flow. Using a regularisation based on the Hodge Laplacian we prove existence and uniqueness of weak solutions for smooth domains. The model problem is then discretised using H(div)-conforming finite element methods, for which we prove error estimates for the velocity approximation in the $L^2$-norm of order $O(h^{k+\frac12})$. We also prove error estimates for the pressure error in the $L^2$-norm.

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