arXiv subjects

Domingo A. Tarzia

Publications and source records attributed to Domingo A. Tarzia.

At least 19 recordsLinked to original sources

arXiv2026

Finite Element Approximation of a Hemivariational Inequality for Steady-State Heat Conduction: Double-Limit Convergence of Penalization and Discretization

In this paper we study the numerical approximation and convergence analysis of a steady-state heat conduction problem with mixed boundary conditions. The physical model is governed by a hemivariational inequality depending on a heat transfer parameter $α> 0$. We consider the finite element approximation of this penalized problem, as well as its corresponding limit problem with a prescribed constant temperature on a part of the boundary. The main theoretical result establishes the strong convergence of the discrete solutions. Specifically, we prove the double-limit convergence of the finite element approximations to the limit solution as the mesh size $h$ tends to zero and the penalization parameter $α$ tends to infinity, independently and simultaneously. We further complement the convergence analysis with an error estimate of optimal order for the discrete limit problem, and we explain why an estimate uniform in the penalization parameter cannot be expected. The theoretical results are illustrated by numerical simulations on four examples verified by the method of manufactured solutions.

math.NA↗

arXiv2026

Explicit Discrete Solution for Some Optimization Problems and Estimations with Respect to the Exact Solution

We consider two steady-state heat conduction systems called, $S$ and $S_α$, in a multidimensional bounded domain $D$ for the Poisson equation with source energy $g$. In one system, we impose mixed boundary conditions (temperature $b$ on the boundary $Γ_1$, heat flux $q$ on $Γ_2$ and an adiabatic condition on $Γ_3$). In the other system, the condition on $Γ_1$ is replaced by a convective heat flux condition with coefficient $α$. For each of these systems, we consider three associated optimization problems $(P_{i})$ and $(P_{iα})$, $i=1,2,3$, where the variable is the source energy $g$, the heat flux $q$ and the environmental temperature $b$, respectively. In the particular case where $D$ is a rectangle, the explicit continuous optimization variables and the corresponding state of the systems are known. In the present work, by using a finite difference scheme, we obtain the discrete systems $({S^h})$ and ${(S^h_α)}$ and discrete optimization problems ${(P^h_i)}$ and ${(P^h_{i α})}$, $i=1,2,3$, where $h$ is the space step in the discretization. Explicit discrete solutions are found, and convergence and estimation errors results are proved when $h$ goes to zero and when $α$ goes to infinity. Moreover, some numerical simulations are provided in order to test theoretical results. Finally, we note that the use of a three-point finite-difference approximation for the Neumann or Robin boundary condition at the boundary improves the global order of convergence from $O(h)$ to $O(h^2)$.

math.NA↗

arXiv2024

A new penalty method for elliptic quasivariational inequalities

We consider a class of elliptic quasivariational inequalities in a reflexive Banach space $X$ for which we recall a convergence criterion obtained in [10]. Each inequality $\cal P$ in the class is governed by a set of constraints $K$ and has a unique solution $u\in K$. The criterion provides necessary and sufficient conditions which guarantee that an arbitrary sequence $\{u_n\}\subset X$ converges to the solution $u$. Then, we consider a sequence $\{\cal P_n\}$ of unconstrained variational-hemivariational inequalities governed by a sequence of parameters $\{λ_n\}\subset\mathbb{R}_+$. We use our criterion to deduce that, if for each $n\in\mathbb{N}$ the term $u_n$ represents a solution of Problem $\cal P_n$, then the sequence $\{u_n\}$ converges to $u$ as $λ_n\to 0$. We apply our abstract results in the study of an elastic frictional contact problem with unilateral constraints and provide the corresponding mechanical interpretations. We also present numerical simulation in the study of a two-dimensional example which represents an evidence of our convergence results.

math.AP↗

arXiv2023

A class of elliptic mixed boundary value problems with $(p,q)$-Laplacian: existence, comparison and optimal control

The paper deals with two nonlinear elliptic equations with $(p,q)$-Laplacian and the Dirichlet-Neumann-Dirichlet (DND) boundary conditions, and Dirich\-let-Neu\-mann-Neumann (DNN) boundary conditions, respectively. Under mild hypotheses, we prove the unique weak solvability of the elliptic mixed boundary value problems. Then, a comparison and a monotonicity results for the solutions of elliptic mixed boundary value problems are established. We examine a convergence result which shows that the solution of (DND) can be approached by the solution of (DNN). Moreover, two optimal control problems governed by (DND) and (DNN), respectively, are considered, and an existence result for optimal control problems is obtained. Finally, we provide a result on asymptotic behavior of optimal controls and system states, when a parameter tends to infinity.

math.AP↗

arXiv2023

A Convergence Criterion for Elliptic Variational Inequalities

We consider an elliptic variational inequality with unilateral constraints in a Hilbert space $X$ which, under appropriate assumptions on the data, has a unique solution $u$. We formulate a convergence criterion to the solution $u$, i.e., we provide necessary and sufficient conditions on a sequence $\{u_n\}\subset X$ which guarantee the convergence $u_n\to u$ in the space $X$. Then, we illustrate the use of this criterion to recover well-known convergence results and well-posedness results in the sense of Tykhonov and Levitin-Polyak. We also provide two applications of our results, in the study of a heat transfer problem and an elastic frictionless contact problem, respectively.

math.AP↗

arXiv2023

Modelling, Analysis and Numerical Simulation of a Spring-Rods System with Unilateral Constraints

In this paper we consider a mathematical model which describes the equilibrium of two elastic rods attached to a nonlinear spring. We derive the variational formulation of the model which is in the form of an elliptic quasivariational inequality for the displacement field. We prove the unique weak solvability of the problem, then we state and prove some convergence results, for which we provide the corresponding mechanical interpretation. Next, we turn to the numerical approximation of the problem based on a finite element scheme. We use a relaxation method to solve the discrete problems that we implement on the computer. Using this method, we provide numerical simulations which validate our convergence results.

math.NA↗

arXiv2023

Numerical analysis of a family of simultaneous distributed-boundary mixed elliptic optimal control problems and their asymptotic behaviour through a commutative diagram and error estimates

In this paper, we consider a family of simultaneous distributed-boundary optimal control problems ($P_α$) on the internal energy and the heat flux for a system governed by a mixed elliptic variational equality with a parameter $α>0$ and a simultaneous distributed-boundary optimal control problem ($P$) governed also by an elliptic variational equality with a Dirichlet boundary condition on the same portion of the boundary. We formulate discrete approximations $\left(P_{h α}\right)$ and $\left(P_h\right)$ of the problems $\left(P_α\right)$ and $(P)$ respectively, for each $h>0$ and for each $α>0$, through the finite element method with Lagrange's triangles of type 1 with parameter $h$ (the longest side of the triangles). The goal of this paper is to study the convergence of this family of discrete simultaneous distributed-boundary mixed elliptic optimal control problems $\left(P_{h α}\right)$ when the parameters $α$ goes to infinity and the parameter $h$ goes to zero simultaneously. We prove the convergence of the problems $\left(P_{h α}\right)$ to the problem $\left(P_h\right)$ when $α\rightarrow +\infty$, for each $h>0$. We study the convergence of the problems $\left(P_{h α}\right)$ and $\left(P_h\right)$, for each $α>0$, when $h \rightarrow 0^+$ obtaining a commutative diagram which relates the continuous and discrete optimal control problems $\left(P_{h α}\right),\left(P_α\right),\left(P_h\right)$ and $(P)$ by taking the limits $h \rightarrow 0^+$ and $α\rightarrow +\infty$ respectively. We also study the double convergence of $\left(P_{h α}\right)$ to $(P)$ when $(h, α) \rightarrow(0^+,+\infty)$ which represents the diagonal convergence in the above commutative diagram.

math.OC↗

arXiv2022

Explicit solutions related to the Rubinstein binary-alloy solidification problem with a heat flux or a convective condition at the fixed face

Similarity solutions for the two-phase Rubinstein binary-alloy solidification problem in a semi-infinite material are developed. These new explicit solutions are obtained by considering two cases: a heat flux or a convective boundary conditions at the fixed face, and the necessary and sufficient conditions on data are also given in order to have an instantaneous solidification process. We also show that all solutions for the binary-alloy solidification problem are equivalent under some restrictions for data. Moreover, this implies that the coefficient which characterizes the solidification front for the Rubinstein solution must verify an inequality as a function of all thermal and boundary conditions.

math.AP↗

arXiv2022

A Class of Moving Boundary Problems with a Source Term. Application of a Reciprocal Transformation

We consider a new Stefan-type problem for the classical heat equation with a latent heat and phase-change temperature depending of the variable time. We prove the equivalence of this Stefan problem with a class of boundary value problems for the nonlinear canonical evolution equation involving a source term with two free boundaries. This equivalence is obtained by applying a reduction to a Burgers equation and a reciprocal-type transformations. Moreover, for a particular case, we obtain a unique explicit solution for the two different problems.

math.AP↗

arXiv2022

Explicit solution for non-classical one-phase Stefan problem with variable thermal coefficients and two different heat source terms

A one-phase Stefan problem for a semi-infinite material is investigated for special functional forms of the thermal conductivity and specific heat depending on the temperature of the phase-change material. Using the similarity transformation technique, an explicit solution for these situations are showed. The mathematical analysis is made for two different kinds of heat source terms, and the existence and uniqueness of the solutions are proved.

math.AP↗

arXiv2021

Heat transfer process with solid-solid interface: Analytical and numerical solutions

This work is aimed at the study and analysis of the heat transport on a metal bar of length $L$ with a solid-solid interface. The process is assumed to be developed along one direction, across two homogeneous and isotropic materials. Analytical and numerical solutions are obtained under continuity conditions at the interface, that is a perfect assembly. The lateral side is assumed to be isolated and a constant thermal source is located at the left-boundary while the right-end stays free allowing the heat to transfer to the surrounding fluid by a convective process. The differences between the analytic solution and temperature measurements at any point on the right would indicate the presence of discontinuities. The greater these differences, the greater the discontinuity in the interface due to thermal resistances, providing a measure of its propagation from the interface and they could be modeled as temperature perturbations. The problem of interest may be described by a parabolic equation with initial, interface and boundary conditions, where the thermal properties, the conductivity and diffusivity coefficients, are piecewise constant functions. The analytic solution is derived by using Fourier methods. Special attention is given to the Sturm-Liouville problem that arises when deriving the solution, since a complicated eigenvalue equation must to be solved. Numerical simulations are conducted by using finite difference schemes where its convergence and stability properties are discussed along with physical interpretations of the results.

math.AP↗

arXiv2021

Distributed optimal control problems for a class of elliptic hemivariational inequalities with a parameter and its asymptotic behavior

In this paper, we study optimal control problems on the internal energy for a system governed by a class of elliptic boundary hemivariational inequalities with a parameter. The system has been originated by a steady-state heat conduction problem with non-monotone multivalued subdifferential boundary condition on a portion of the boundary of the domain described by the Clarke generalized gradient of a locally Lipschitz function. We prove an existence result for the optimal controls and we show an asymptotic result for the optimal controls and the system states, when the parameter, like a heat transfer coefficient, tends to infinity on a portion of the boundary.

math.OC↗

arXiv2021

Existence, comparison, and convergence results for a class of elliptic hemivariational inequalities

In this paper we study a class of elliptic boundary hemivariational inequalities which originates in the steady-state heat conduction problem with nonmonotone multivalued subdifferential boundary condition on a portion of the boundary described by the Clarke generalized gradient of a locally Lipschitz function. First, we prove a new existence result for the inequality employing the theory of pseudomonotone operators. Next, we give a result on comparison of solutions, and provide sufficient conditions that guarantee the asymptotic behavior of solution, when the heat transfer coefficient tends to infinity. Further, we show a result on the continuous dependence of solution on the internal energy and heat flux. Finally, some examples of convex and nonconvex potentials illustrate our hypotheses.

math.AP↗

arXiv2021

About the Convergence of a Family of Initial Boundary Value Problems for a Fractional Diffusion Equation with Robin Conditions

We consider a family of initial boundary value problems governed by a fractional diffusion equation with Caputo derivative in time, where the parameter is the Newton heat transfer coefficient linked to the Robin condition on the boundary. For each problem we prove existence and uniqueness of solution by a Fourier approach. This will enable us to also prove the convergence of the family of solutions to the solution of the limit problem, which is obtained by replacing the Robin boundary condition with a Dirichlet boundary condition.

math.AP↗

arXiv2021

Tykhonov Well-posedness of a Heat Transfer Problem with Unilateral Constraints

We consider an elliptic boundary value problem with unilateral constraints and subdifferential boundary conditions. The problem describes the heat transfer in a domain $D\subset\R^d$ and its weak formulation is in the form of a hemivariational inequality for the temperature field, denoted by $\cP$. We associate to Problem $\cP$ an optimal control problem, denoted by $\cQ$. Then, using appropriate Tykhonov triples, governed by a nonlinear operator $G$ and a convex $\wK$, we provide results concerning the well-posedness of problems $\cP$ and $\cQ$. Our main results are Theorems 14 and 18, together with their corollaries. Their proofs are based on arguments of compactness, lower semicontinuity and pseudomonotonicity. Moreover, we consider three relevant perturbations of the heat transfer boundary valued problem which lead to penalty versions of Problem $\cP$, constructed with particular choices of $G$ and $\wK$. We prove that Theorems 14 and 18 as well as their corollaries can be applied in the study of these problems, in order to obtain various convergence results.

math.AP↗

arXiv2020

Optimal control of differential quasivariational inequalities with applications in contact mechanics

We consider a differential quasivariational inequality for which we state and prove the continuous dependence of the solution with respect to the data. This convergence result allows us to prove the existence of at least one optimal pair for an associated control problem. Finally, we illustrate our abstract results in the study of a free boundary problem which describes the equilibrium of a viscoelastic body in frictionless contact with a foundation made of a rigid body coveblack by a rigid-elastic layer.

math.AP↗

arXiv2020

On the Tykhonov Well-posedness of an Antiplane Shear Problem

We consider a boundary value problem which describes the frictional antiplane shear of an elastic body. The process is static and friction is modeled with a slip-dependent version of Coulomb's law of dry friction. The weak formulation of the problem is in the form of a quasivariational inequality for the displacement field, denoted by $\cP$. We associated to problem $\cP$ a boundary optimal control problem, denoted by $\cQ$. For Problem $\cP$ we introduce the concept of well-posedness and for Problem $\cQ$ we introduce the concept of weakly and weakly generalized well-posedness, both associated to appropriate Tykhonov triples. Our main result are Theorems \ref{t1} and \ref{t2}. Theorem \ref{t1} provides the well-posedness of Problem $\cP$ and, as a consequence, the continuous dependence of the solution with respect to the data. Theorem \ref{t2} provides the weakly generalized well-posedness of Problem $\cQ$ and, under additional hypothesis, its weakly well posedness. The proofs of these theorems are based on arguments of compactness, lower semicontinuity, monotonicity and various estimates. Moreover, we provide the mechanical interpretation of our well-posedness results.

math.AP↗

arXiv2020

Approximate solutions to one-phase Stefan-like problems with space-dependent latent heat

The work in this paper concerns the study of different approximations for one-dimensional one-phase Stefan-like problems with a space-dependent latent heat. It is considered two different problems, which differ from each other in their boundary condition imposed at the fixed face: Dirichlet and Robin conditions. The approximate solutions are obtained by applying the heat balance integral method (HBIM), a modified heat balance integral method, the refined integral method (RIM) . Taking advantage of the exact analytical solutions we compare and test the accuracy of the approximate solutions. The analysis is carried out using the dimensionless generalized Stefan number (Ste) and Biot number (Bi). It is also studied the case when Bi goes to infinity in the problem with a convective condition, recovering the approximate solutions when a temperature condition is imposed at the fixed face. Some numerical simulations are provided in order to assert which of the approximate integral methods turns out to be optimal. Moreover, we pose an approximate technique based on minimizing the least-squares error, obtaining also approximate solutions for the classical Stefan problem.

math.AP↗