arXiv subjects

Gabriel R. Barrenechea

Publications and source records attributed to Gabriel R. Barrenechea.

At least 19 recordsLinked to original sources

arXiv2026

Analysis and approximation of a two-dimensional induction heating problem

In this paper, we analyse the existence of solutions and finite element approximation of a steady-state two-dimensional induction heating problem. One of the main difficulties of the problem is its right-hand side which, at a first sight, is only integrable. Using a priori regularity results for the PDEs involved it is shown that the natural weak formulation of the problem can be justified. Then, we study the finite element approximation and prove that the standard Galerkin FEM converges in convex domains and under suitable conditions on the mesh. We improve on this result by applying the recently-proposed bound-preserving method (BPM) to the heat equation, and show that this method converges to a solution of the problem under less stringent conditions on the domain and the mesh. As these analyses are carried out without any assumption on regularity of the solutions, then the convergence of the finite element method also proves existence of solutions. Several numerical experiments confirm the theoretical results, and showcase the improvement provided by the use of the bound-preserving method over the standard finite element method.

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arXiv2026

An Aubin-Nitsche Lemma for a positivity-preserving finite element method

In this work we prove an Aubin-Nitsche Lemma for a positivity-preserving discretisation of an elliptic problem. Due to the nonlinearity of the discretisation, the result requires as a first step the proposal of a linearised adjoint problem that can be linked to the method of choice by appropriately selecting weights. This linearised adjoint problem, together with a regularity result, allow the proof of an optimal-order error estimate in the $L^2$-norm of the error.

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arXiv2026

A finite element method preserving the eigenvalue range of symmetric tensor fields

This paper presents a finite element method that preserves (at the degrees of freedom) the eigenvalue range of the solution of tensor-valued time-dependent convection--diffusion equations. Starting from a high-order spatial baseline discretisation (in this case, the CIP stabilised finite element method), our approach formulates the fully discrete problem as a variational inequality posed on a closed convex set of tensor-valued functions that respect the same eigenvalue bounds at their degrees of freedom. The numerical realisation of the scheme relies on the definition of a projection that, at each node, performs the diagonalisation of the tensor and then truncates the eigenvalues to lie within the prescribed bounds. The temporal discretisation is carried out using the implicit Euler method, and unconditional stability and optimal-order error estimates are proven for this choice. Numerical experiments confirm the theoretical findings and illustrate the method's ability to maintain eigenvalue constraints while accurately approximating solutions in the convection-dominated regime.

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arXiv2025

A bound-preserving and conservative enriched Galerkin method for elliptic problems

We propose a locally conservative enriched Galerkin scheme that preserves the physical bounds for an elliptic problem. To this end, we use a substantial over-penalization of the discrete solution's jumps to obtain optimal convergence. To avoid the ill-conditioning issues that arise in over-penalized schemes, we introduce an involved splitting approach that separates the system of equations for the discontinuous solution part from the system of equations for the continuous solution part, yielding well-behaved subproblems. We prove the existence of discrete solutions and optimal error estimates, which are validated numerically.

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arXiv2025

A finite element method for a non-Newtonian dilute polymer fluid

We study the discretisation of a uniaxial (rank-one) reduction of the Oldroyd-B model for dilute polymer solutions, in which the conformation tensor is represented as $\sig = \vec b \otimes \vec b$. Building on structural analogies with MHD, we formulate a finite element framework compatible with the de Rham complex, so that the discrete velocity is exactly divergence-free. The spatial discretisation combines an interior-penalty treatment of viscosity with upwind transport to control stress layers and we prove inf-sup conditions on the mixed pairs. For time-stepping, we design an IMEX scheme that is linear at each step and show well-posedness of the fully discrete problem together with a discrete energy law mirroring the continuum dissipation. Numerical experiments on canonical benchmarks (lid-driven cavity, pipe-with-cavity and $4{:}1$ planar contraction) demonstrate accuracy and robustness for moderate-to-high Weissenberg numbers, capturing sharp stress gradients and corner singularities while retaining the efficiency gains of the uniaxial model. The results indicate that de Rham-compatible discretisations coupled with energy-stable IMEX time integration provide a reliable pathway for viscoelastic computations at elevated elasticity.

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arXiv2025

A nodally bound-preserving composite discontinuous Galerkin method on polytopic meshes

We introduce a nodally bound-preserving Galerkin method for second-order elliptic problems on general polygonal/polyhedral, henceforth collectively termed as \emph{polytopic}, meshes. Starting from an interior penalty discontinuous Galerkin (DG) formulation posed on a polytopic mesh, the method enforces preservation of \emph{a priori} prescribed upper and lower bounds for the numerical solution at an arbitrary number of user-defined points \emph{within} each polytopic element. This is achieved by employing a simplicial submesh and enforcing bound preservation at the submesh nodes via a nonlinear iteration. By construction, the submeshing procedure preserves the order of accuracy of the DG method, \emph{without} introducing any additional global numerical degrees of freedom compared to the baseline DG method, thereby, falling into the category of composite finite element approaches. A salient feature of the proposed method is that it automatically reverts to the standard DG method on polytopic meshes when no prescribed bound violation occurs. In particular, the choice of the discontinuity-penalisation parameter is independent of the submesh granularity. The resulting composite method combines the geometric flexibility of polytopic meshes with the accuracy and stability of discontinuous Galerkin discretisations, while rigorously guaranteeing bound preservation. The existence and uniqueness of the numerical solution is proven. A priori error bounds, assuming sufficient regularity of the exact solution are shown, employing a non-standard construction of discrete nodally bound-preserving interpolant. Numerical experiments confirm optimal convergence for smooth problems and demonstrate robustness in the presence of sharp gradients, such as boundary and interior layers.

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arXiv2025

A nodally bound-preserving discontinuous Galerkin method for the drift-diffusion equation

In this work, we introduce and analyse discontinuous Galerkin (dG) methods for the drift-diffusion model. We explore two dG formulations: a classical interior penalty approach and a nodally bound-preserving method. Whilst the interior penalty method demonstrates well-posedness and convergence, it fails to guarantee non-negativity of the solution. To address this deficit, which is often important to ensure in applications, we employ a positivity-preserving method based on a convex subset formulation, ensuring the non-negativity of the solution at the Lagrange nodes. We validate our findings by summarising extensive numerical experiments, highlighting the novelty and effectiveness of our approach in handling the complexities of charge carrier transport.

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arXiv2025

A nodally bound-preserving finite element method for time-dependent convection-diffusion equations

This paper presents a new method to approximate the time-dependent convection-diffusion equations using conforming finite element methods, ensuring that the discrete solution respects the physical bounds imposed by the differential equation. The method is built by defining, at each time step, a convex set of admissible finite element functions (that is, the ones that satisfy the global bounds at their degrees of freedom) and seeks for a discrete solution in this admissible set. A family of $θ$-schemes is used as time integrators, and well-posedness of the discrete schemes is proven for the whole family, but stability and optimal-order error estimates are proven for the implicit Euler scheme. Nevertheless, our numerical experiments show that the method also provides stable and optimally-convergent solutions when the Crank-Nicolson method is used.

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arXiv2023

A nodally bound-preserving finite element method for reaction-convection-diffusion equations

This paper introduces a novel approach to approximate a broad range of reaction-convection-diffusion equations using conforming finite element methods while providing a discrete solution respecting the physical bounds given by the underlying differential equation. The main result of this work demonstrates that the numerical solution achieves accuracy of $O(h^k)$ in the energy norm, where $k$ represents the underlying polynomial degree. To validate the approach, a series of numerical experiments is conducted for various problem instances. Comparisons with the linear continuous interior penalty stabilised method, and the algebraic flux-correction scheme (for the piecewise linear finite element case) have been carried out, where we can observe the favourable performance of the current approach.

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arXiv2023

Implicit-explicit schemes for incompressible flow problems with variable viscosity

In this work we study different Implicit-Explicit (IMEX) schemes for incompressible flow problems with variable viscosity. Unlike most previous work on IMEX schemes, which focuses on the convective part, we here focus on treating parts of the diffusive term explicitly to reduce the coupling between the velocity components. We present different, both monolithic and fractional-step, IMEX alternatives for the variable-viscosity Navier--Stokes system, analysing their theoretical and algorithmic properties. Stability results are proven for all the methods presented, with all these results being unconditional, except for one of the discretisations using a fractional-step scheme, where a CFL condition (in terms of the problem data) is required for showing stability. Our analysis is supported by a series of numerical experiments.

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arXiv2023

Finite element methods respecting the discrete maximum principle for convection-diffusion equations

Convection-diffusion-reaction equations model the conservation of scalar quantities. From the analytic point of view, solution of these equations satisfy under certain conditions maximum principles, which represent physical bounds of the solution. That the same bounds are respected by numerical approximations of the solution is often of utmost importance in practice. The mathematical formulation of this property, which contributes to the physical consistency of a method, is called Discrete Maximum Principle (DMP). In many applications, convection dominates diffusion by several orders of magnitude. It is well known that standard discretizations typically do not satisfy the DMP in this convection-dominated regime. In fact, in this case, it turns out to be a challenging problem to construct discretizations that, on the one hand, respect the DMP and, on the other hand, compute accurate solutions. This paper presents a survey on finite element methods, with a main focus on the convection-dominated regime, that satisfy a local or a global DMP. The concepts of the underlying numerical analysis are discussed. The survey reveals that for the steady-state problem there are only a few discretizations, all of them nonlinear, that at the same time satisfy the DMP and compute reasonably accurate solutions, e.g., algebraically stabilized schemes. Moreover, most of these discretizations have been developed in recent years, showing the enormous progress that has been achieved lately. Methods based on algebraic stabilization, nonlinear and linear ones, are currently as well the only finite element methods that combine the satisfaction of the global DMP and accurate numerical results for the evolutionary equations in the convection-dominated situation.

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arXiv2022

Continuous Interior Penalty stabilization for divergence-free finite element methods

In this paper we propose, analyze, and test numerically a pressure-robust stabilized finite element for a linearized problem in incompressible fluid mechanics, namely, the steady Oseen equation with low viscosity. Stabilization terms are defined by jumps of different combinations of derivatives for the convective term over the element faces of the triangulation of the domain. With the help of these stabilizing terms, and the fact the finite element space is assumed to provide a point-wise divergence-free velocity, an $\mathcal O\big(h^{k+\frac12}\big)$ error estimate in the $L^2$-norm is proved for the method (in the convection-dominated regime), and optimal order estimates in the remaining norms of the error. Numerical results supporting the theoretical findings are provided.

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arXiv2021

Analysis of a stabilised finite element method for power-law fluids

A low-order finite element method is constructed and analysed for an incompressible non-Newtonian flow problem with power-law rheology. The method is based on a continuous piecewise linear approximation of the velocity field and piecewise constant approximation of the pressure. Stabilisation, in the form of pressure jumps, is added to the formulation to compensate for the failure of the inf-sup condition, and using an appropriate lifting of the pressure jumps a divergence-free approximation to the velocity field is built and included in the discretisation of the convection term. This construction allows us to prove the convergence of the resulting finite element method for the entire range $r>\frac{2 d}{d+2}$ of the power-law index $r$ for which weak solutions to the model are known to exist in $d$ space dimensions, $d \in \{2,3\}$.

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arXiv2020

A pressure-robust discretization of Oseen's equation using stabilization in the vorticity equation

Discretization of Navier-Stokes' equations using pressure-robust finite element methods is considered for the high Reynolds number regime. To counter oscillations due to dominating convection we add a stabilization based on a bulk term in the form of a residual-based least squares stabilization of the vorticity equation supplemented by a penalty term on (certain components of) the gradient jump over the elements faces. Since the stabilization is based on the vorticity equation, it is independent of the pressure gradients, which makes it pressure-robust. Thus, we prove pressure-independent error estimates in the linearized case, known as Oseen's problem. In fact, we prove an $O(h^{k+\frac12})$ error estimate in the $L^2$-norm that is known to be the best that can be expected for this type of problem. Numerical examples are provided that, in addition to confirming the theoretical results, show that the present method compares favorably to the classical residual-based SUPG stabilization.

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arXiv2018

Stabilised hybrid discontinuous Galerkin methods for the Stokes problem with non-standard boundary conditions

In several studies it has been observed that, when using stabilised $\mathbb{P}_k^{}\times\mathbb{P}_k^{}$ elements for both velocity and pressure, the error for the pressure is smaller, or even of a higher order in some cases, than the one obtained when using inf-sup stable $\mathbb{P}_k^{}\times\mathbb{P}_{k-1}^{}$ (although no formal proof of either of these facts has been given). This increase in polynomial order requires the introduction of stabilising terms, since the finite element pairs used do not stability the inf-sup condition. With this motivation, we apply the stabilisation approach to the hybrid discontinuous Galerkin discretisation for the Stokes problem with non-standard boundary conditions.

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arXiv2018

The inf-sup stability of the lowest order Taylor-Hood pair on Anisotropic Meshes

Uniform LBB conditions are desirable to approximate the solution of Navier-Stokes, Oseen, and Stokes equations on anisotropic meshes and to enable anisotropic refinements. We prove such conditions for the second order Taylor-Hood pairs $\mathbb{Q}_2 \times \mathbb{Q}_1$ and $\mathbb{P}_2 \times \mathbb{P}_1$ on a class of anisotropic meshes. These meshes may contain refined edge and corner patches. To this end, we generalise Verfürth's trick and recent results by some of the authors.

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arXiv2017

Hybrid discontinuous Galerkin discretisation and domain decomposition preconditioners for the Stokes problem

Solving the Stokes equation by an optimal domain decomposition method derived algebraically involves the use of non standard interface conditions whose discretisation is not trivial. For this reason the use of approximation methods such as hybrid discontinuous Galerkin appears as an appropriate strategy: on the one hand they provide the best compromise in terms of the number of degrees of freedom in between standard continuous and discontinuous Galerkin methods, and on the other hand the degrees of freedom used in the non standard interface conditions are naturally defined at the boundary between elements. In this paper we introduce the coupling between a well chosen discretisation method (hybrid discontinuous Galerkin) and a novel and efficient domain decomposition method to solve the Stokes system. We present the detailed analysis of the hybrid discontinuous Galerkin method for the Stokes problem with non standard boundary conditions. This analysis is supported by numerical evidence. In addition, the advantage of the new preconditioners over more classical choices is also supported by numerical experiments.

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arXiv2017

Numerical assessment of two-level domain decomposition preconditioners for incompressible Stokes and elasticity equations

Solving the linear elasticity and Stokes equations by an optimal domain decomposition method derived algebraically involves the use of non standard interface conditions. The one-level domain decomposition preconditioners are based on the solution of local problems. This has the undesired consequence that the results are not scalable, it means that the number of iterations needed to reach convergence increases with the number of subdomains. This is the reason why in this work we introduce, and test numerically, two-level preconditioners. Such preconditioners use a coarse space in their construction. We consider the nearly incompressible elasticity problems and Stokes equations, and discretise them by using two finite element methods, namely, the hybrid discontinuous Galerkin and Taylor-Hood discretisations.

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