Explanipedia

Asymptotic compatibility of parametrized optimal design problems Open

Tadele Mengesha, Abner J. Salgado, Joshua M. Siktar · 2025

We study optimal design problems where the design corresponds to a coefficient in the principal part of the state equation. The state equation, in addition, is parameter dependent, and we allow it to change type in the limit of this (model…

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

Gabriel R. Barrenechea, Abner J. Salgado · 2025

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-p…

Finite element discretization of weighted $Φ$-Laplace problems Open

Enrique Otárola, Abner J. Salgado · 2024

We study the finite element approximation of problems involving the weighted $Φ$-Laplacian, where $Φ$ is an $N$-function and the weight belongs to the class $A_Φ$. In particular, we consider a boundary value problem and an obstacle problem…

Asymptotic compatibility of parametrized optimal design problems Open

Tadele Mengesha, Abner J. Salgado, Joshua M. Siktar · 2024

We study optimal design problems where the design corresponds to a coefficient in the principal part of the state equation. The state equation, in addition, is parameter dependent, and we allow it to change type in the limit of this (model…

A Semi-Analytic Diagonalization FEM for the Spectral Fractional Laplacian Open

Abner J. Salgado, Steven F. Sawyer · 2024

We present a technique for approximating solutions to the spectral fractional Laplacian, which is based on the Caffarelli-Silvestre extension and diagonalization. Our scheme uses the analytic solution to the associated eigenvalue problem i…

Analysis and finite element approximation of a diffuse interface approach to the Stokes--Biot coupling Open

Francis R. A. Aznaran, Martina Bukač, Boris Muha, Abner J. Salgado · 2024

We consider the interaction between a poroelastic structure, described using the Biot model in primal form, and a free-flowing fluid, modelled with the time-dependent incompressible Stokes equations. We propose a diffuse interface model in…

Analysis and approximation of elliptic problems with Uhlenbeck structure in convex polytopes Open

Tadele Mengesha, Enrique Otárola, Abner J. Salgado · 2024

We prove the well posedness in weighted Sobolev spaces of certain linear and nonlinear elliptic boundary value problems posed on convex domains and under singular forcing. It is assumed that the weights belong to the Muckenhoupt class $A_p…

Monotone two-scale methods for a class of integrodifferential operators and applications Open

Juan Pablo Borthagaray, Ricardo H. Nochetto, Abner J. Salgado, Céline Torres · 2024

We develop a monotone, two-scale discretization for a class of integrodifferential operators of order $2s$, $s \in (0,1)$. We apply it to develop numerical schemes, and derive pointwise convergence rates, for linear and obstacle problems g…

Numerical approximation of variational problems with orthotropic growth Open

Anna Kh. Balci, Lars Diening, Abner J. Salgado · 2023

We consider the numerical approximation of variational problems with orthotropic growth, that is those where the integrand depends strongly on the coordinate directions with possibly different growth in each direction. Under realistic regu…

The linear elasticity system under singular forces Open

Alejandro Allendes, Gilberto Campaña, Enrique Otárola, Abner J. Salgado · 2023

We study the linear elasticity system subject to singular forces. We show existence and uniqueness of solutions in two frameworks: weighted Sobolev spaces, where the weight belongs to the Muckenhoupt class $A_2$; and standard Sobolev space…

Pointwise gradient estimate of the ritz projection Open

Lars Diening, Julian Rolfes, Abner J. Salgado · 2023

Let $Ω\subset \mathbb{R}^n$ be a convex polytope ($n \leq 3$). The Ritz projection is the best approximation, in the $W^{1,2}_0$-norm, to a given function in a finite element space. When such finite element spaces are constructed on the ba…

On the Optimal Control of a Linear Peridynamics Model Open

Tadele Mengesha, Abner J. Salgado, Joshua M. Siktar · 2023

We study a non-local optimal control problem involving a linear, bond-based peridynamics model. In addition to existence and uniqueness of solutions to our problem, we investigate their behavior as the horizon parameter $δ$, which controls…

Time fractional gradient flows: Theory and numerics Open

Wenbo Li, Abner J. Salgado · 2023

We develop the theory of fractional gradient flows: an evolution aimed at the minimization of a convex, lower semicontinuous energy, with memory effects. This memory is characterized by the fact that the negative of the (sub)gradient of th…

Convergent, with rates, methods for normalized infinity Laplace, and related, equations Open

Wenbo Li, Abner J. Salgado · 2022

We propose a monotone, and consistent numerical scheme for the approximation of the Dirichlet problem for the normalized Infinity Laplacian, which could be related to the family of so--called two--scale methods. We show that this method is…

Asymptotic preserving methods for fluid electron-fluid models in the large magnetic field limit with mathematically guaranteed properties (Final Report) Open

Ignacio Tomaš, John N. Shadid, Matthias Maier, Abner J. Salgado · 2022

The current manuscript is a final report on the activities carried out under the Project LDRD-CIS #226834. In scientific terms, the work reported in this manuscript is a continuation of the efforts started with Project LDRD-express #223796…

Diagonally implicit Runge-Kutta schemes: Discrete energy-balance laws and compactness properties Open

Abner J. Salgado, Ignacio Tomaš · 2022

We study diagonally implicit Runge-Kutta (DIRK) schemes when applied to abstract evolution problems that fit into the Gelfand-triple framework. We introduce novel stability notions that are well-suited to this setting and provide simple, n…

Benchmark computations of the phase field crystal and functionalized Cahn-Hilliard equations via fully implicit, Nesterov accelerated schemes Open

Jea-Hyun Park, Abner J. Salgado, Steven P. Wise · 2022

We introduce a fast solver for the phase field crystal (PFC) and functionalized Cahn-Hilliard (FCH) equations with periodic boundary conditions on a rectangular domain that features the preconditioned Nesterov accelerated gradient descent …

Preconditioned Accelerated Gradient Descent Methods for Locally Lipschitz Smooth Objectives with Applications to the Solution of Nonlinear PDEs Open

Jea-Hyun Park, Abner J. Salgado, Steven M. Wise · 2021

The Darcy problem with porosity depending exponentially on the pressure Open

Zerihun Kinfe Birhanu, Tadele Mengesha, Abner J. Salgado · 2021

Estimation of the continuity constants for Bogovskiĭ and regularized Poincaré integral operators Open

Johnny Guzmán, Abner J. Salgado · 2021

The stationary Boussinesq problem under singular forcing Open

Alejandro Allendes, Enrique Otárola, Abner J. Salgado · 2021

In Lipschitz two- and three-dimensional domains, we study the existence for the so-called Boussinesq model of thermally driven convection under singular forcing. By singular we mean that the heat source is allowed to belong to [Formula: se…

Estimation of the continuity constants for Bogovski\u{\i} and regularized Poincar\'e integral operators Open

Johnny Guzmán, Abner J. Salgado · 2020

We study the dependence of the continuity constants for the regularized Poincar\'e and Bogovski\u{\i} integral operators acting on differential forms defined on a domain $\Omega$ of $\mathbb{R}^n$. We, in particular, study the dependence o…

Estimation of the continuity constants for Bogovski\uı and regularized Poincaré integral operators Open

Johnny Guzmán, Abner J. Salgado · 2020

We study the dependence of the continuity constants for the regularized Poincaré and Bogovski\uı integral operators acting on differential forms defined on a domain $Ω$ of $\mathbb{R}^n$. We, in particular, study the dependence of such con…

Preconditioned accelerated gradient descent methods for locally\n Lipschitz smooth objectives with applications to the solution of nonlinear\n PDEs Open

Jea-Hyun Park, Abner J. Salgado, Steven M. Wise · 2020

We develop a theoretical foundation for the application of Nesterov's\naccelerated gradient descent method (AGD) to the approximation of solutions of\na wide class of partial differential equations (PDEs). This is achieved by\nproving the …

On the analysis and approximation of some models of fluids over weighted spaces on convex polyhedra Open

Enrique Otárola, Abner J. Salgado · 2020

We study the Stokes problem over convex polyhedral domains on weighted Sobolev spaces. The weight is assumed to belong to the Muckenhoupt class $A_q$ for $q \in (1,\infty)$. We show that the Stokes problem is well-posed for all $q$. In add…

A Posteriori Error Estimates for the Stationary Navier--Stokes Equations with Dirac Measures Open

Alejandro Allendes, Enrique Otárola, Abner J. Salgado · 2020

In two dimensions, we propose and analyze an a posteriori error estimator for finite element approximations of the stationary Navier Stokes equations with singular sources on Lipschitz, but not necessarily convex, polygonal domains. Under …

Stability of the Stokes projection on weighted spaces and applications Open

Ricardo G. Durán, Enrique Otárola, Abner J. Salgado · 2019

We show that, on convex polytopes and two or three dimensions, the finite element Stokes projection is stable on weighted spaces $\mathbf{W}^{1,p}_0(ω,Ω) \times L^p(ω,Ω)$, where the weight belongs to a certain Muckenhoupt class and the int…

Weighted Sobolev regularity and rate of approximation of the obstacle problem for the integral fractional Laplacian Open

Juan Pablo Borthagaray, Ricardo H. Nochetto, Abner J. Salgado · 2019

We obtain regularity results in weighted Sobolev spaces for the solution of the obstacle problem for the integral fractional Laplacian [Formula: see text] in a Lipschitz bounded domain [Formula: see text] satisfying the exterior ball condi…

A posteriori error estimates for the stationary Navier Stokes equations\n with Dirac measures Open

Alejandro Allendes, Enrique Otárola, Abner J. Salgado · 2019

In two dimensions, we propose and analyze an a posteriori error estimator for\nfinite element approximations of the stationary Navier Stokes equations with\nsingular sources on Lipschitz, but not necessarily convex, polygonal domains.\nUnd…

Finite element approximation of an obstacle problem for a class of integro–differential operators Open

Andrea Bonito, Wenyu Lei, Abner J. Salgado · 2019

We study the regularity of the solution to an obstacle problem for a class of integro–differential operators. The differential part is a second order elliptic operator, whereas the nonlocal part is given by the integral fractional Laplacia…

Abner J. Salgado YOU? Author Swipe