Domingo Tarzia

Domingo Tarzia
Austral University (Argentina) · Matematica

Habilitation, Univ. Paris VI
Research in Free Boundary Problems (Stefan problem) and Vicerrector Investigacion Universidad Austral

About

293
Publications
32,032
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2,261
Citations
Citations since 2016
88 Research Items
1055 Citations
2016201720182019202020212022050100150200
2016201720182019202020212022050100150200
2016201720182019202020212022050100150200
2016201720182019202020212022050100150200
Additional affiliations
February 1991 - present
Austral University (Argentina)
Position
  • Head of Department
November 1983 - present
National Scientific and Technical Research Council
Position
  • Investigador Superior
August 1979 - December 1990
Rosario National University
Position
  • Professor (Full)

Publications

Publications (293)
Article
Full-text available
A one-phase Stefan problem for a semi-infinite material is studied 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 exact solution for these situations are shown. The mathematical analysis is made for two different...
Article
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 al...
Article
Full-text available
The paper deals with two nonlinear elliptic equations with (p, q)-Laplacian and the Dirichlet–Neumann–Dirichlet (DND) boundary conditions, and Dirichlet–Neumann–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 monotoni...
Preprint
Full-text available
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...
Preprint
Full-text available
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 equivalenc...
Article
In this work, a bar fully insulated on its lateral surface. composed by two different unknown materials is considered. For the analytical solution, it is assumed a perfectly assembly solid-solid interface, so no heat loss due to friction is present. This is an ideal scenario, so this loss and possible measurement errors are included by simulating n...
Article
Full-text available
In this paper we obtain self-similarity solutions for a one-phase one-dimensional fractional space Stefan problem in terms of the three parametric Mittag-Leffler function Eα,m,l(z). We consider Dirichlet and Neumann conditions at the fixed face, involving Caputo fractional space derivatives of order 0<α<1. We recover the solution for the classical...
Preprint
Full-text available
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 d...
Article
Full-text available
The goal of this paper is to investigate a new class of elliptic mixed boundary value problems involving a nonlinear and nonhomogeneous partial differential operator [Formula: see text]-Laplacian, and a multivalued term represented by Clarke’s generalized gradient. First, we apply a surjectivity result for multivalued pseudomonotone operators to ex...
Article
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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 resul...
Preprint
Full-text available
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 asse...
Preprint
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 dom...
Article
Full-text available
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 assemb...
Article
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 dom...
Preprint
Full-text available
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 resul...
Article
Full-text available
We consider a heat conduction problem S with mixed boundary conditions in an n -dimensional domain Ω with regular boundary and a family of problems S α with also mixed boundary conditions in Ω, where α > 0 is the heat transfer coefficient on the portion of the boundary Γ ¹ . In relation to these state systems, we formulate Neumann boundary optimal...
Preprint
Full-text available
An inverse problem for a stationary heat transfer process is studied for a totally isolated bar on its lateral surface, made up of two consecutive sections of different, isotropic and homogeneous materials, perfectly assembly, where one of the materials, that is unreachable and unknown, has to be identified. The length of the bar is assumed to be m...
Preprint
Full-text available
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 al...
Article
Full-text available
An inverse problem for a stationary heat transfer process is studied for a totally isolated bar on its lateral surface, made up of two consecutive sections of different, isotropic and homogeneous materials, perfectly assembly, where one of the materials, that is unreachable and unknown, has to be identified. The length of the bar is assumed to be m...
Preprint
Full-text available
An inverse problem for a stationary heat transfer process is studied for a totally isolated bar on its lateral surface, of negligible diameter, made up of two consecutive sections of different, isotropic and homogeneous materials. At the left boundary, a Dirichlet type condition is imposed that represents a constant temperature source while a Robin...
Preprint
We consider a heat conduction problem $S$ with mixed boundary conditions in a $n$-dimensional domain $\Omega$ with regular boundary and a family of problems $S_{\alpha}$ with also mixed boundary conditions in $\Omega$, where $\alpha>0$ is the heat transfer coefficient on the portion of the boundary $\Gamma_{1}$. In relation to these state systems,...
Preprint
Full-text available
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 p...
Article
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 p...
Article
Full-text available
An inverse problem for a stationary heat transfer process is studied for a totally isolated bar on its lateral surface, of negligible diameter, made up of two consecutive sections of different, isotropic and homogeneous materials. At the left boundary, a Dirichlet type condition is imposed that represents a constant temperature source while a Robin...
Preprint
Full-text available
In this paper we obtain self-similarity solutions for a one-phase one-dimensional fractional space one-phase Stefan problem in terms of the three parametric Mittag-Leffer function $E_{\alpha,m;l}(z)$. We consider Dirichlet and Newmann conditions at the fixed face, involving Caputo fractional space derivatives of order $0 < \alpha < 1$. We recover t...
Preprint
Full-text available
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 p...
Preprint
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 ass...
Article
Full-text available
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 \({{{\mathcal...
Preprint
Full-text available
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 obtain...
Article
Full-text available
We consider an optimal control problem Q governed by an elliptic quasivariational inequality with unilateral constraints. We associate to Q a new optimal control problem Q˜, obtained by perturbing the state inequality (including the set of constraints and the nonlinear operator) and the cost functional, as well. Then, we provide sufficient conditio...
Article
Full-text available
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 obtain...
Preprint
Full-text available
We consider an optimal control problem $\cQ$ governed by an elliptic quasivariational inequality with unilateral constraints. The existence of optimal pairs of the problem is a well known result, see \cite{SS}, for instance. We associate to $\cQ$ a new optimal control problem $\wQ$, obtained by perturbing the state inequality (including the set of...
Article
Full-text available
We consider a steady-state heat conduction problem in a multidimensional bounded domain \(\Omega \) for the Poisson equation with constant internal energy g and mixed boundary conditions given by a constant temperature b in the portion \(\Gamma _1\) of the boundary and a constant heat flux q in the remaining portion \(\Gamma _2\) of the boundary. M...
Article
Two fractional two-phase Stefan-like problems are considered by using Riemann-Liouville and Caputo derivatives of order α ∈ (0, 1) verifying that they coincide with the same classical Stefan problem at the limit case when α=1. For both problems, explicit solutions in terms of the Wright functions are presented. Even though the similarity of the two...
Article
One dimensional Stefan problems for a semi-infinite material with temperature dependent thermal coefficients are considered. Existence and uniqueness of solution are obtained imposing a Dirichlet, a Neumann or a Robin type condition at fixed face x=0. Moreover, it is proved that the solution of the problem with the Robin type condition converges to...
Preprint
Full-text available
Two fractional two-phase Stefan-like problems are considered by using Riemann-Liouville and Caputo derivatives of order $\alpha \in (0, 1)$ verifying that they coincide with the same classical Stefan problem at the limit case when $\alpha=1$. For both problems, explicit solutions in terms of the Wright functions are presented. Even though the simil...
Article
Full-text available
We address the existence and uniqueness of the so-called modified error function that arises in the study of phase-change problems with specific heat and thermal conductivity given by linear functions of the material temperature. This function is defined from a differential problem that depends on two parameters which are closely related with the s...
Preprint
Full-text available
One-dimensional free boundary problem for a nonlinear diffusion - convection equation with a Dirichlet condition at fixed face $x=0$, variable in time, is considered. Throught several transformations the problem is reduced to a free boundary problem for a diffusion equation and the integral formulation is obtained. By using fixed point theorems, th...
Preprint
Full-text available
We consider a heat conduction problem $S$ with mixed boundary conditions in a n-dimensional domain $\Omega$ with regular boundary $\Gamma$ and a family of problems $S_{\alpha}$, where the parameter $\alpha>0$ is the heat transfer coefficient on the portion of the boundary $\Gamma_{1}$. In relation to these state systems, we formulate simultaneous \...
Article
One-dimensional free boundary problem for a nonlinear diffusion–convection equation with a Dirichlet condition at fixed face x=0, variable in time, is considered. Through several transformations the problem is reduced to a free boundary problem for a diffusion equation and the integral formulation is obtained. By using fixed point theorems, the exi...
Article
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On the paper D. Burini, S De Lillo, G. Fioriti, Acta Mech., 229 No. 10 (2018), pp 4215–4228. It is a letter to the Editor (3 pages).
Preprint
Full-text available
We address the existence and uniqueness of the so-called modified error function that arises in the study of phase-change problems with specific heat and thermal conductivity given by linear functions of the material temperature. This function is defined from a differential problem that depends on two parameters which are closely related with the s...
Preprint
Full-text available
Motivated by the modeling of temperature regulation in some mediums, we consider the non-classical heat conduction equation in the domain $D=\mathbb{R}^{n-1}\times\br^{+}$ for which the internal energy supply depends on an average in the time variable of the heat flux $(y, s)\mapsto V(y,s)= u_{x}(0 , y , s)$ on the boundary $S=\partial D$. The solu...
Preprint
Full-text available
One dimensional Stefan problems for a semi-infinite material with temperature dependent thermal coefficients are considered. Existence and uniqueness of solution are obtained imposing a Dirichlet or a Robin type condition at fixed face $x=0$. Moreover, it is proved that the solution of the problem with the Robin type condition converges to the solu...
Preprint
Full-text available
We consider a steady-state heat conduction problem in a multidimensional bounded domain Omega for the Poisson equation with constant internal energy g and mixed boundary conditions given by a constant temperature b in the portion Gamma1 of the boundary and a constant heat flux q in the remaining portion Gamma2 of the boundary. Moreover, we consider...
Preprint
Full-text available
In this chapter we consider different approximations for the one-dimensional one-phase Stefan problem corresponding to the fusion process of a semi-infinite material with a temperature boundary condition at the fixed face and non-linear temperature-dependent thermal conductivity. The knowledge of the exact solution of this problem, allows to compar...
Preprint
Full-text available
In this paper we consider two different Stefan problems for a semi-infinite material for the non classical heat equation with a source which depends on the heat flux at the fixed face x = 0. One of them (with constant temperature on x = 0) was studied in [4] where it was found a unique exact solution of similarity type and the other (with a convect...
Preprint
In this paper, the p-generalized modified error function is defined as the solution to a non-linear ordinary differential problem of second order with a Robin type condition at $x=0$. Existence and uniqueness of a non-negative analytic solution is proved by using a fixed point strategy. It is shown that the p-generalized modified error function con...
Preprint
Full-text available
Recently it was obtained in [Tarzia, Thermal Sci. 21A (2017) 1-11] for the classical two-phase Lam\'e-Clapeyron-Stefan problem an equivalence between the temperature and convective boundary conditions at the fixed face under a certain restriction. Motivated by this article we study the two-phase Stefan problem for a semi-infinite material with a la...
Preprint
Full-text available
This paper deals with the fractional Caputo--Fabrizio derivative and some basic properties related. A computation of this fractional derivative to power functions is given in terms of Mittag--Lefler functions. The inverse operator named the fractional Integral of Caputo--Fabrizio is also analyzed. The main result consists in the proof of existence...
Article
In this paper we consider a one-dimensional one-phase Stefan problem corresponding to the solidification process of a semi-infinite material with a convective boundary condition at the fixed face. The exact solution of this problem, available recently in the literature, enable us to test the accuracy of the approximate solutions obtained by applyin...
Preprint
Full-text available
In this paper we consider a one-dimensional one-phase Stefan problem corresponding to the solidification process of a semi-infinite material with a convective boundary condition at the fixed face. The exact solution of this problem, available recently in the literature, enable us to test the accuracy of the approximate solutions obtained by applyin...
Article
Full-text available
A one-phase Stefan-type problem for a semi-infinite material which has as its main feature a variable latent heat that depends on the power of the position and the velocity of the moving boundary is studied. Exact solutions of similarity type are obtained for the cases when Neumann or Robin boundary conditions are imposed at the fixed face. Require...
Article
Full-text available
In this article, we obtain explicit approximations of the modified error function introduced in Cho, Sunderland. Journal of Heat Transfer 96-2 (1974), 214-217, as part of a Stefan problem with a temperature-dependent thermal conductivity. This function depends on a parameter δ, which is related to the thermal conductivity in the original phase-chan...
Preprint
Full-text available
A mathematical model for a one-phase change problem (particularly a Stefan problem) with a memory flux, is obtained. The hypothesis that the weighted sum of fluxes back in time is proportional to the gradient of temperature is considered. The model obtained involves fractional derivatives with respect on time in the sense of Caputo and in the sense...
Preprint
Full-text available
A generalized Neumann solution for the two-phase fractional Lam\'e--Clapeyron--Stefan problem for a semi--infinite material with constant initial temperature and a particular heat flux condition at the fixed face is obtained, when a restriction on data is satisfied. The fractional derivative in the Caputo sense of order $\al \in (0,1)$ respect on t...
Article
Full-text available
A two-phase solidification process for a one-dimensional semi-infinite material is considered. It is assumed that it is ensued from a constant bulk temperature present in the vicinity of the fixed boundary, which it is modelled through a convective condition (Robin condition). The interface between the two phases is idealized as a mushy region and...
Article
Full-text available
In this article it is proved the existence of similarity solutions for a one-phase Stefan problem with temperature-dependent thermal conductivity and a Robin condition at the fixed face. The temperature distribution is obtained through a generalized modified error function which is defined as the solution to a nonlinear ordinary differential proble...
Article
Full-text available
We establish in this paper the equivalence between a Volterra integral equation of second kind and a singular ordinary differential equation of third order with two initial conditions and an integral boundary condition, with a real parameter. This equivalence allow us to obtain the solution to some problems for nonclassical heat equation, the conti...
Article
Full-text available
Recently, in Tarzia (Thermal Sci 21A:1–11, 2017) for the classical two-phase Lamé–Clapeyron–Stefan problem an equivalence between the temperature and convective boundary conditions at the fixed face under a certain restriction was obtained. Motivated by this article we study the two-phase Stefan problem for a semi-infinite material with a latent he...
Article
Full-text available
A generalized Neumann solution for the two-phase fractional Lamé–Clapeyron–Stefan problem for a semi-infinite material with constant initial temperature and a particular heat flux condition at the fixed face is obtained, when a restriction on data is satisfied. The fractional derivative in the Caputo sense of order \(\alpha \in (0,1)\) respect on t...
Article
Full-text available
We consider two different Stefan problems for a semi-infinite material for the nonclassical heat equation with a source that depends on the heat flux at the fixed face. One of them, with constant temperature at the fixed face, was already studied in literature and the other, with a convective boundary condition at the fixed face, is presented in th...
Article
Full-text available
Two fractional Stefan problems are considered by using Riemann-Liouville and Caputo derivatives of order $\alpha \in (0,1)$ such that in the limit case ($\alpha =1$) both problems coincide with the same classical Stefan problem. For the one and the other problem, explicit solutions in terms of the Wright functions are presented. We prove that these...
Data
In this article it is proved the existence of similarity solutions for a one-phase Stefan problem with temperature-dependent thermal conductivity and a Robin condition at the fixed face. The temperature distribution is obtained through a generalized modified error function which is defined as the solution to a nonlinear ordinary differential proble...
Poster
Full-text available
Expansion of the Bindeman-Melnik model for zircon growth and dissolution in magma
Article
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A generalized cumulative uptake formula of nutrient uptake by roots following our previous formula (Reginato-Tarzia, Comm. Soil Sci. and Plant., 33 (2002), 821-830) is developed. Cumulative nutrient uptake obtained by this formula is compared with the simulated results obtained by the Claassen and Barber (Claassen and Barber, Agronomy J., 68 (1976)...
Article
Full-text available
In this work, is studied comparatively a unidimensional model of transport and simultaneous intake of nutrients and water by crop roots. The model is derived from the Barber-Cushman model with a mobile boundary modification (time-varying domain). In the equations of this model the transport of nutrients is coupled to the transport of water, which fo...
Preprint
Full-text available
In this article, we obtain explicit approximations of the modified error function introduced in Cho, Sunderland. Journal of Heat Transfer 96-2 (1974), 214-217, as part of a Stefan problem with a temperature-dependent thermal conductivity. This function depends on a parameter δ, which is related to the thermal conductivity in the original phase-chan...
Conference Paper
Full-text available
Resumen: La función de error modificada, dependiente de un parámetro conocido δ ≥ −1, definida por Cho y Sunderland en J. Heat Trasfer, 96C(1974), 214-217, fue utilizada para resolver numerosos problemas de cambio de fase. En este trabajo se presenta una representación de la función de error modificada como una serie de potencias en el parámetro δ...
Conference Paper
Full-text available
Resumen: En este trabajo se considera un problema de solidificación a dos fases para un material unidimensional semi-infinito. Se supone que el proceso de cambio de fase se inicia a partir de la presencia de una temperatura constante en las cercanías de la frontera fija, lo cual se modela con una condición convectiva (condición de Robin). La interf...
Conference Paper
Full-text available
Resumen: En este trabajo se considera un material de cambio de fase unidimensional semi-infinito con coeficientes térmicos constantes entre los cuales son desconocidos simultáneamente el calor específico y la conductividad térmica. Con el objetivo de determinar estos coeficientes, se supone que el material se encuentra bajo un proceso de fusión con...
Article
Full-text available
We consider the non-classical heat conduction equation, in the domain $D=\br^{n-1}\times\br^{+}$, for which the internal energy supply depends on an integral function in the time variable of % $(y , t)\mapsto \int_{0}^{t} u_{x}(0 , y , s) ds$, %where $u_{x}(0 , y , s)$ is the heat flux on the boundary $S=\partial D$, with homogeneous Dirichlet boun...
Article
Full-text available
We consider a semi-infinite one-dimensional phase-change material with two unknown constant thermal coefficients among the latent heat per unit mass, the specific heat, the mass density and the thermal conductivity. Aiming at the determination of them, we consider an inverse one-phase Stefan problem with an over-specified condition at the fixed bou...
Article
Full-text available
From the one-dimensional consolidation of fine-grained soils with threshold gradient, it can be derived a special type of Stefan problems where the seepage front, because of the presence of this threshold gradient, exhibits the features of a moving boundary. In this type of problems, in contrast with the classical Stefan problem, the latent heat is...
Article
This article is devoted to prove the existence and uniqueness of solution to the non-linear second order differential problem through which is defined the modified error function introduced in Cho-Sunderland, J. Heat Transfer, 96-2:214-217, 1974. We prove here that there exists a unique non-negative analytic solution for small positive values of th...
Article
Full-text available
An explicit solution of a similarity type is obtained for a one-phase Stefan problem in a semi-infinite material using Kummer functions. Motivated by [D.A. Tarzia, Relationship between Neumann solutions for two phase Lam\'e-Clapeyron-Stefan problems with convective and temperature boundary conditions, Thermal Sci.(2016) DOI 10.2298/TSCI 140607003T,...
Article
Full-text available
A one-dimensional fractional one-phase Stefan problem with a temperature boundary condition at the fixed face is considered by using the Riemann–Liouville derivative. This formulation is more convenient than the one given in Roscani and Santillan ( Fract. Calc. Appl. Anal. , 16 , No 4 (2013), 802–815) and Tarzia and Ceretani ( Fract. Calc. Appl. An...
Article
Full-text available
The numerical analysis of a family of distributed mixed optimal control problems governed by elliptic variational inequalities (with parameter $\alpha >0$) is obtained through the finite element method when its parameter $h\rightarrow 0$. We also obtain the limit of the discrete optimal control and the associated state system solutions when $\alpha...
Article
Full-text available
We consider a simple investment project with the following parameters: I>0: Initial investment which is amortizable in n years; n: Number of years the investment allows production with constant output per year; A>0: Annual amortization (A=I/n); Q>0: Quantity of products sold per year; Cv>0: Variable cost per unit; p>0: Price of the product with p>C...
Conference Paper
The convergence of a family of continuous distributed mixed elliptic optimal control problems (\( P_{\alpha } \)), governed by elliptic variational equalities, when the parameter \( \alpha \to \infty \) was studied in Gariboldi - Tarzia, Appl. Math. Optim., 47 (2003), 213-230 and it has been proved that it is convergent to a distributed mixed ellip...
Article
Full-text available
We have considered the simultaneous determination of two unknown thermal coefficients for a semi-infinite material which was under a phase-change process with a mushy zone according to the model of Solomon, Wilson and Alexiades. It was assumed that the material was initially liquid at its melting temperature and that the solidification process bega...
Article
Full-text available
The numerical analysis of a family of distributed mixed optimal control problems governed by elliptic variational inequalities (with parameter $\alpha >0$) is obtained through the finite element method when its parameter $h\rightarrow 0$. We also obtain the limit of the discrete optimal control and the associated state system solutions when $\alpha...