Abdallah Bradji - Academia.edu (original) (raw)

Papers by Abdallah Bradji

Research paper thumbnail of A New Gradient Scheme of a Time Fractional Fokker–Planck Equation with Time Independent Forcing and Its Convergence Analysis

Springer proceedings in mathematics & statistics, 2020

We apply the GDM (Gradient Discretization Method) developed recently in [5, 6] to approximate the... more We apply the GDM (Gradient Discretization Method) developed recently in [5, 6] to approximate the time fractional Fokker–Planck equation with time independent forcing in any space dimension. Using [5] which dealt with GDM for linear advection problems, we develop a new fully discrete implicit GS (Gradient Scheme) for the stated model. We prove new discrete a priori estimates which yield estimates on the discrete solution in \(L^\infty (L^2)\) and \(L^2(H^1)\) discrete norms. Thanks to these discrete a priori estimates, we prove new error estimates in the discrete norms of \(L^\infty (L^2)\) and \(L^2(H^1)\). The main ingredients in the proof of these error estimates are the use of the stated discrete a priori estimates and a comparison with some well chosen auxiliary schemes. These auxiliary schemes are approximations of convective-diffusive elliptic problems in each time level. We state without proof the convergence analysis of these auxiliary schemes. Such proof uses some adaptations of the [6, Proof of Theorem 2.28] dealt with GDM for the case of elliptic diffusion problems. These results hold for all the schemes within the framework of GDM. This work can be viewed as an extension to our recent one [2].

Research paper thumbnail of A New Optimal <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow></mrow><annotation encoding="application/x-tex"></annotation></semantics></math></span><span class="katex-html" aria-hidden="true"></span></span>L^{\infty }(H^1)$$–Error Estimate of a SUSHI Scheme for the Time Fractional Diffusion Equation

Springer proceedings in mathematics & statistics, 2020

We consider a finite volume scheme, using the general mesh of [8], for the TFDE (time fractional ... more We consider a finite volume scheme, using the general mesh of [8], for the TFDE (time fractional diffusion equation) in any space dimension. The time discretization is performed using a uniform mesh. We prove a new discrete \(L^\infty (H^1)\)–a priori estimate. Such a priori estimate is proved thanks to the use of the new tool of the discrete Laplace operator developed recently in [7]. Thanks to this a priori estimate, we prove a new optimal convergence order in the discrete \(L^\infty (H^1)\)–norm. These results improve the ones of [1, 4] which dealt respectively with finite volume and GDM (Gradient Discretization Method) for the TFDE. In [4], we only proved a priori estimate and error estimate in the discrete \(L^\infty (L^2)\)–norm whereas in [1] we proved a priori estimate and error estimate in the discrete \(L^2(H^1)\)–norm. The a priori estimate as well as the error estimate presented here were stated without proof for the first time in [3, Remark 1, p. 443] in the context of the general framework of GDM and [2, Remark 1, p. 205] in the context of finite volume methods. They also were mentioned, as future works, in [1, Remark 4.1].

Research paper thumbnail of Steady-state inhomogeneous diffusion with generalized oblique boundary conditions

ESAIM, Jul 26, 2023

We consider the elliptic diffusion (steady-state heat conduction) equation with spacedependent co... more We consider the elliptic diffusion (steady-state heat conduction) equation with spacedependent conductivity and inhomogeneous source subject to a generalized oblique boundary condition on a part of the boundary and Dirichlet or Neumann boundary conditions on the remaining part. The oblique boundary condition represents a linear combination between the dependent variable and its normal and tangential derivatives at the boundary. We first prove the well-posedness of the continuous problems. We then develop new finite volume schemes for these problems and prove rigorously the stability and convergence of these schemes. We also address an application to the inverse corrosion problem concerning the reconstruction of the coefficients present in the generalized oblique boundary condition that is prescribed over a portion Γ0 of the boundary Ω from Cauchy data on the complementary portion Γ1 = Ω∖Γ0.

Research paper thumbnail of SUSHI for a Bingham Flow Type Problem

Lecture Notes in Computer Science, 2023

Research paper thumbnail of Convergence Analysis of a Finite Volume Scheme for a Distributed Order Diffusion Equation

Lecture Notes in Computer Science, 2023

Research paper thumbnail of SUSHI for a Non-linear Time Fractional Diffusion Equation with a Time Independent Delay

Lecture Notes in Computer Science, 2023

Research paper thumbnail of Some Abstract Error Estimates of a Finite Volume Scheme for the Wave Equation on General Nonconforming Multidimensional Spatial Meshes

Springer proceedings in mathematics, 2011

Bradji Finite volume for the wave equation tu-logo ur-logo Aim... Motivation Problem to be discre... more Bradji Finite volume for the wave equation tu-logo ur-logo Aim... Motivation Problem to be discretized Mesh A first result: discretization scheme A second main result: error estimates Conclusion Some paths to be followed

Research paper thumbnail of Some Convergence Results of a Multi-dimensional Finite Volume Scheme for a Time-Fractional Diffusion-Wave Equation

Springer proceedings in mathematics & statistics, 2017

We present an implicit finite volume scheme for a linear time-fractional diffusion-wave equation ... more We present an implicit finite volume scheme for a linear time-fractional diffusion-wave equation using the discrete gradient introduced in Eymard et al. (IMA J Numer Anal 30:1009–1043, 2010, [2]). A convergence order for the error between the gradient of the exact solution and the discrete gradient of the approximate solution is proved. This yields an \(L^\infty (L^2)\)–error estimate.

Research paper thumbnail of Finite Volumes for Complex Applications X—Volume 1, Elliptic and Parabolic Problems

Springer Proceedings in Mathematics & Statistics

Research paper thumbnail of Rational Approximation Preconditioners for Multiphysics Problems

Numerical Methods and Applications

We consider a class of mathematical models describing multiphysics phenomena interacting through ... more We consider a class of mathematical models describing multiphysics phenomena interacting through interfaces. On such interfaces, the traces of the fields lie (approximately) in the range of a weighted sum of two fractional differential operators. We use a rational function approximation to precondition such operators. We first demonstrate the robustness of the approximation for ordinary functions given by weighted sums of fractional exponents. Additionally, we present more realistic examples utilizing the proposed preconditioning techniques in interface coupling between Darcy and Stokes equations.

Research paper thumbnail of Novel analysis approach for the convergence of a second order time accurate mixed finite element scheme for parabolic equations

Computers & Mathematics with Applications

Research paper thumbnail of A Finite Volume Scheme for a Wave Equation with Several Time Independent Delays

Large-Scale Scientific Computing, 2022

Research paper thumbnail of Note on the Convergence of a Finite Volume Scheme for a Second Order Hyperbolic Equation with a Time Delay in Any Space Dimension

Finite Volumes for Complex Applications IX - Methods, Theoretical Aspects, Examples, 2020

In this note, we establish a finite volume scheme for a model of a second order hyperbolic equati... more In this note, we establish a finite volume scheme for a model of a second order hyperbolic equation with a time delay in any space dimension. This model is considered in [10, 11] where some exponential stability estimates and oscillatory behaviour are proved. The scheme we shall present uses, as space discretization, the general class of nonconforming finite volume meshes of [5]. In addition to the proof of the existence and uniqueness of the discrete solution, we develop a new discrete a priori estimate. Thanks to this a priori estimate, we prove error estimates in discrete seminorms of \(L^\infty (H^1_0)\), \(L^\infty (L^2)\), and \(W^{1,\infty }(L^2)\). This work can be viewed as extension to the previous ones [2, 4] which dealt with the analysis of finite volume methods for respectively semilinear parabolic equations with a time delay and the wave equation.

Research paper thumbnail of A New Gradient Scheme of a Time Fractional Fokker–Planck Equation with Time Independent Forcing and Its Convergence Analysis

We apply the GDM (Gradient Discretization Method) developed recently in [5, 6] to approximate the... more We apply the GDM (Gradient Discretization Method) developed recently in [5, 6] to approximate the time fractional Fokker–Planck equation with time independent forcing in any space dimension. Using [5] which dealt with GDM for linear advection problems, we develop a new fully discrete implicit GS (Gradient Scheme) for the stated model. We prove new discrete a priori estimates which yield estimates on the discrete solution in \(L^\infty (L^2)\) and \(L^2(H^1)\) discrete norms. Thanks to these discrete a priori estimates, we prove new error estimates in the discrete norms of \(L^\infty (L^2)\) and \(L^2(H^1)\). The main ingredients in the proof of these error estimates are the use of the stated discrete a priori estimates and a comparison with some well chosen auxiliary schemes. These auxiliary schemes are approximations of convective-diffusive elliptic problems in each time level. We state without proof the convergence analysis of these auxiliary schemes. Such proof uses some adaptati...

Research paper thumbnail of On the Convergence Rate of the Dirichlet-Neumann Iteration for Coupled Poisson Problems on Unstructured Grids

Finite Volumes for Complex Applications IX - Methods, Theoretical Aspects, Examples, 2020

We consider thermal fluid structure interaction with a partitioned approach, where typically, a f... more We consider thermal fluid structure interaction with a partitioned approach, where typically, a finite volume and a finite element code would be coupled. As a model problem, we consider two coupled Poisson problems with heat conductivities λ 1 , λ 2 in one dimension on intervals of length l 1 and l 2. Hereby, we consider linear discretizations on arbitrary meshes, such as finite volumes, finite differences, finite elements. For these, we prove that the convergence rate of the Dirichlet-Neumann iteration is given by λ 1 l 2 /λ 2 l 1 and is thus independent of discretization and mesh.

Research paper thumbnail of A new error estimate for a fully finite element discretization scheme for parabolic equations using Crank-Nicolson method

Mathematica Bohemica, 2014

Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents ... more Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.

Research paper thumbnail of A new analysis for the convergence of the gradient discretization method for multidimensional time fractional diffusion and diffusion-wave equations

Computers & Mathematics with Applications, 2019

Research paper thumbnail of On the convergence and convergence order of finite volume gradient schemes for oblique derivative boundary value problems

Computational and Applied Mathematics, 2017

This work is an improvement of the previous note (Bradji in: Fuhrmann et al., Finite volumes for ... more This work is an improvement of the previous note (Bradji in: Fuhrmann et al., Finite volumes for complex applications VII-methods, theoretical aspects. Proceedings of the FVCA 7, Berlin, 2014) which dealt with the convergence analysis of a finite volume scheme for the Poisson's equation with a linear oblique derivative boundary condition. The formulation of the finite volume scheme given in Bradji (in: Fuhrmann et al., Finite volumes for complex applications VII-methods, theoretical aspects. Proceedings of the FVCA 7, Berlin, 2014) involves the discrete gradient introduced recently in Eymard et al. (IMA J Numer Anal 30(4):1009-1043, 2010). In this paper, we consider the convergence analysis of finite volume schemes involving the discrete gradient of Eymard et al. (IMA J Numer Anal 30(4):1009-1043, 2010) for elliptic and parabolic equations with linear oblique derivative boundary conditions. Linear oblique derivative boundary conditions arise for instance in the study of the motion of water in a canal, cf. Lesnic (Commun Numer Methods Eng 23(12):1071-1080, 2007). We derive error estimates in several norms which allow us to get error estimates for the approximations of the exact solutions and its first derivatives. In particular, we provide an error estimate between the gradient of the exact solutions and the discrete gradient of the approximate solutions. Convergence of the family of finite volume approximate solutions towards the exact solution under weak regularity assumption is also investigated. In the case of parabolic equations with oblique derivative boundary conditions, we develop a new discrete a priori estimate result. The proof of this result is based on the use of a discrete mean Poincaré-Wirtinger inequality. Thanks to the stated a priori estimate Communicated by Raphaèle Herbin.

Research paper thumbnail of An analysis for the convergence order of gradient schemes for semilinear parabolic equations

Computers & Mathematics with Applications, 2016

Gradient schemes are numerical methods, which can be conforming and nonconforming, have been rece... more Gradient schemes are numerical methods, which can be conforming and nonconforming, have been recently developed in Droniou et al. (2013), Droniou et al. (2015), Eymard et al. (2012) and references therein to approximate different types of partial differential equations. They are written in a discrete variational formulation and based on the approximation of functions and gradients. The aim of the present paper is to provide gradient schemes along with an analysis for the convergence order of these schemes for semilinear parabolic equations in any space dimension. We present three gradient schemes. The first two schemes are nonlinear whereas the third one is linear. The existence and uniqueness of the discrete solutions for the first two schemes is proved, thanks to the use of the method of contractive mapping, under the assumption that the mesh size of the time discretization k is small, whereas the existence and uniqueness of the discrete solution for the third scheme is proved for arbitrary k. We provide a convergence rate analysis in discrete semi-norms of L ∞ (H 1) and W 1,2 (L 2) and in the norm of L ∞ (L 2). We prove that the order in space is the same one proved in Eymard et al. (2012) when approximating elliptic equations and one or two in time. The existence, uniqueness, and the convergence results stated above do not require any relation between spacial and temporal discretizations. As an application of these results, we focus on the gradient schemes which use the discrete gradient introduced recently in the SUSHI method (Eymard et al., 2010) and we provide some numerical tests.

Research paper thumbnail of On the discretization of ohmic losses

We consider a nonlinear system of elliptic equations, which arises when modelling the heat diffus... more We consider a nonlinear system of elliptic equations, which arises when modelling the heat diffusion problem coupled with the electrical diffusion problem. The ohmic losses which appear as a source term in the heat diffusion equation yield a nonlinear term which lies in L 1 . A finite element scheme is considered for the discretization of the system; we show that the approximate solution obtained with the scheme converges, up to a subsequence, to a solution of the coupled elliptic system. RÉSUMÉ. On considère ici un système d'équations elliptiques non linéaires, qui provient de la modé-lisation d'un problème de diffusion de la chaleur couplé à un problème de diffusion électrique. L'effet Joule se modélise par un terme non linéaire qui est dans L 1 . On étudie un schéma éléments finis pour la discrétisation de ce système. On montre que la solutions approchée obtenue, converge, à une sous-suite près, vers une solution du problème continu.

Research paper thumbnail of A New Gradient Scheme of a Time Fractional Fokker–Planck Equation with Time Independent Forcing and Its Convergence Analysis

Springer proceedings in mathematics & statistics, 2020

We apply the GDM (Gradient Discretization Method) developed recently in [5, 6] to approximate the... more We apply the GDM (Gradient Discretization Method) developed recently in [5, 6] to approximate the time fractional Fokker–Planck equation with time independent forcing in any space dimension. Using [5] which dealt with GDM for linear advection problems, we develop a new fully discrete implicit GS (Gradient Scheme) for the stated model. We prove new discrete a priori estimates which yield estimates on the discrete solution in \(L^\infty (L^2)\) and \(L^2(H^1)\) discrete norms. Thanks to these discrete a priori estimates, we prove new error estimates in the discrete norms of \(L^\infty (L^2)\) and \(L^2(H^1)\). The main ingredients in the proof of these error estimates are the use of the stated discrete a priori estimates and a comparison with some well chosen auxiliary schemes. These auxiliary schemes are approximations of convective-diffusive elliptic problems in each time level. We state without proof the convergence analysis of these auxiliary schemes. Such proof uses some adaptations of the [6, Proof of Theorem 2.28] dealt with GDM for the case of elliptic diffusion problems. These results hold for all the schemes within the framework of GDM. This work can be viewed as an extension to our recent one [2].

Research paper thumbnail of A New Optimal <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow></mrow><annotation encoding="application/x-tex"></annotation></semantics></math></span><span class="katex-html" aria-hidden="true"></span></span>L^{\infty }(H^1)$$–Error Estimate of a SUSHI Scheme for the Time Fractional Diffusion Equation

Springer proceedings in mathematics & statistics, 2020

We consider a finite volume scheme, using the general mesh of [8], for the TFDE (time fractional ... more We consider a finite volume scheme, using the general mesh of [8], for the TFDE (time fractional diffusion equation) in any space dimension. The time discretization is performed using a uniform mesh. We prove a new discrete \(L^\infty (H^1)\)–a priori estimate. Such a priori estimate is proved thanks to the use of the new tool of the discrete Laplace operator developed recently in [7]. Thanks to this a priori estimate, we prove a new optimal convergence order in the discrete \(L^\infty (H^1)\)–norm. These results improve the ones of [1, 4] which dealt respectively with finite volume and GDM (Gradient Discretization Method) for the TFDE. In [4], we only proved a priori estimate and error estimate in the discrete \(L^\infty (L^2)\)–norm whereas in [1] we proved a priori estimate and error estimate in the discrete \(L^2(H^1)\)–norm. The a priori estimate as well as the error estimate presented here were stated without proof for the first time in [3, Remark 1, p. 443] in the context of the general framework of GDM and [2, Remark 1, p. 205] in the context of finite volume methods. They also were mentioned, as future works, in [1, Remark 4.1].

Research paper thumbnail of Steady-state inhomogeneous diffusion with generalized oblique boundary conditions

ESAIM, Jul 26, 2023

We consider the elliptic diffusion (steady-state heat conduction) equation with spacedependent co... more We consider the elliptic diffusion (steady-state heat conduction) equation with spacedependent conductivity and inhomogeneous source subject to a generalized oblique boundary condition on a part of the boundary and Dirichlet or Neumann boundary conditions on the remaining part. The oblique boundary condition represents a linear combination between the dependent variable and its normal and tangential derivatives at the boundary. We first prove the well-posedness of the continuous problems. We then develop new finite volume schemes for these problems and prove rigorously the stability and convergence of these schemes. We also address an application to the inverse corrosion problem concerning the reconstruction of the coefficients present in the generalized oblique boundary condition that is prescribed over a portion Γ0 of the boundary Ω from Cauchy data on the complementary portion Γ1 = Ω∖Γ0.

Research paper thumbnail of SUSHI for a Bingham Flow Type Problem

Lecture Notes in Computer Science, 2023

Research paper thumbnail of Convergence Analysis of a Finite Volume Scheme for a Distributed Order Diffusion Equation

Lecture Notes in Computer Science, 2023

Research paper thumbnail of SUSHI for a Non-linear Time Fractional Diffusion Equation with a Time Independent Delay

Lecture Notes in Computer Science, 2023

Research paper thumbnail of Some Abstract Error Estimates of a Finite Volume Scheme for the Wave Equation on General Nonconforming Multidimensional Spatial Meshes

Springer proceedings in mathematics, 2011

Bradji Finite volume for the wave equation tu-logo ur-logo Aim... Motivation Problem to be discre... more Bradji Finite volume for the wave equation tu-logo ur-logo Aim... Motivation Problem to be discretized Mesh A first result: discretization scheme A second main result: error estimates Conclusion Some paths to be followed

Research paper thumbnail of Some Convergence Results of a Multi-dimensional Finite Volume Scheme for a Time-Fractional Diffusion-Wave Equation

Springer proceedings in mathematics & statistics, 2017

We present an implicit finite volume scheme for a linear time-fractional diffusion-wave equation ... more We present an implicit finite volume scheme for a linear time-fractional diffusion-wave equation using the discrete gradient introduced in Eymard et al. (IMA J Numer Anal 30:1009–1043, 2010, [2]). A convergence order for the error between the gradient of the exact solution and the discrete gradient of the approximate solution is proved. This yields an \(L^\infty (L^2)\)–error estimate.

Research paper thumbnail of Finite Volumes for Complex Applications X—Volume 1, Elliptic and Parabolic Problems

Springer Proceedings in Mathematics & Statistics

Research paper thumbnail of Rational Approximation Preconditioners for Multiphysics Problems

Numerical Methods and Applications

We consider a class of mathematical models describing multiphysics phenomena interacting through ... more We consider a class of mathematical models describing multiphysics phenomena interacting through interfaces. On such interfaces, the traces of the fields lie (approximately) in the range of a weighted sum of two fractional differential operators. We use a rational function approximation to precondition such operators. We first demonstrate the robustness of the approximation for ordinary functions given by weighted sums of fractional exponents. Additionally, we present more realistic examples utilizing the proposed preconditioning techniques in interface coupling between Darcy and Stokes equations.

Research paper thumbnail of Novel analysis approach for the convergence of a second order time accurate mixed finite element scheme for parabolic equations

Computers & Mathematics with Applications

Research paper thumbnail of A Finite Volume Scheme for a Wave Equation with Several Time Independent Delays

Large-Scale Scientific Computing, 2022

Research paper thumbnail of Note on the Convergence of a Finite Volume Scheme for a Second Order Hyperbolic Equation with a Time Delay in Any Space Dimension

Finite Volumes for Complex Applications IX - Methods, Theoretical Aspects, Examples, 2020

In this note, we establish a finite volume scheme for a model of a second order hyperbolic equati... more In this note, we establish a finite volume scheme for a model of a second order hyperbolic equation with a time delay in any space dimension. This model is considered in [10, 11] where some exponential stability estimates and oscillatory behaviour are proved. The scheme we shall present uses, as space discretization, the general class of nonconforming finite volume meshes of [5]. In addition to the proof of the existence and uniqueness of the discrete solution, we develop a new discrete a priori estimate. Thanks to this a priori estimate, we prove error estimates in discrete seminorms of \(L^\infty (H^1_0)\), \(L^\infty (L^2)\), and \(W^{1,\infty }(L^2)\). This work can be viewed as extension to the previous ones [2, 4] which dealt with the analysis of finite volume methods for respectively semilinear parabolic equations with a time delay and the wave equation.

Research paper thumbnail of A New Gradient Scheme of a Time Fractional Fokker–Planck Equation with Time Independent Forcing and Its Convergence Analysis

We apply the GDM (Gradient Discretization Method) developed recently in [5, 6] to approximate the... more We apply the GDM (Gradient Discretization Method) developed recently in [5, 6] to approximate the time fractional Fokker–Planck equation with time independent forcing in any space dimension. Using [5] which dealt with GDM for linear advection problems, we develop a new fully discrete implicit GS (Gradient Scheme) for the stated model. We prove new discrete a priori estimates which yield estimates on the discrete solution in \(L^\infty (L^2)\) and \(L^2(H^1)\) discrete norms. Thanks to these discrete a priori estimates, we prove new error estimates in the discrete norms of \(L^\infty (L^2)\) and \(L^2(H^1)\). The main ingredients in the proof of these error estimates are the use of the stated discrete a priori estimates and a comparison with some well chosen auxiliary schemes. These auxiliary schemes are approximations of convective-diffusive elliptic problems in each time level. We state without proof the convergence analysis of these auxiliary schemes. Such proof uses some adaptati...

Research paper thumbnail of On the Convergence Rate of the Dirichlet-Neumann Iteration for Coupled Poisson Problems on Unstructured Grids

Finite Volumes for Complex Applications IX - Methods, Theoretical Aspects, Examples, 2020

We consider thermal fluid structure interaction with a partitioned approach, where typically, a f... more We consider thermal fluid structure interaction with a partitioned approach, where typically, a finite volume and a finite element code would be coupled. As a model problem, we consider two coupled Poisson problems with heat conductivities λ 1 , λ 2 in one dimension on intervals of length l 1 and l 2. Hereby, we consider linear discretizations on arbitrary meshes, such as finite volumes, finite differences, finite elements. For these, we prove that the convergence rate of the Dirichlet-Neumann iteration is given by λ 1 l 2 /λ 2 l 1 and is thus independent of discretization and mesh.

Research paper thumbnail of A new error estimate for a fully finite element discretization scheme for parabolic equations using Crank-Nicolson method

Mathematica Bohemica, 2014

Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents ... more Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.

Research paper thumbnail of A new analysis for the convergence of the gradient discretization method for multidimensional time fractional diffusion and diffusion-wave equations

Computers & Mathematics with Applications, 2019

Research paper thumbnail of On the convergence and convergence order of finite volume gradient schemes for oblique derivative boundary value problems

Computational and Applied Mathematics, 2017

This work is an improvement of the previous note (Bradji in: Fuhrmann et al., Finite volumes for ... more This work is an improvement of the previous note (Bradji in: Fuhrmann et al., Finite volumes for complex applications VII-methods, theoretical aspects. Proceedings of the FVCA 7, Berlin, 2014) which dealt with the convergence analysis of a finite volume scheme for the Poisson's equation with a linear oblique derivative boundary condition. The formulation of the finite volume scheme given in Bradji (in: Fuhrmann et al., Finite volumes for complex applications VII-methods, theoretical aspects. Proceedings of the FVCA 7, Berlin, 2014) involves the discrete gradient introduced recently in Eymard et al. (IMA J Numer Anal 30(4):1009-1043, 2010). In this paper, we consider the convergence analysis of finite volume schemes involving the discrete gradient of Eymard et al. (IMA J Numer Anal 30(4):1009-1043, 2010) for elliptic and parabolic equations with linear oblique derivative boundary conditions. Linear oblique derivative boundary conditions arise for instance in the study of the motion of water in a canal, cf. Lesnic (Commun Numer Methods Eng 23(12):1071-1080, 2007). We derive error estimates in several norms which allow us to get error estimates for the approximations of the exact solutions and its first derivatives. In particular, we provide an error estimate between the gradient of the exact solutions and the discrete gradient of the approximate solutions. Convergence of the family of finite volume approximate solutions towards the exact solution under weak regularity assumption is also investigated. In the case of parabolic equations with oblique derivative boundary conditions, we develop a new discrete a priori estimate result. The proof of this result is based on the use of a discrete mean Poincaré-Wirtinger inequality. Thanks to the stated a priori estimate Communicated by Raphaèle Herbin.

Research paper thumbnail of An analysis for the convergence order of gradient schemes for semilinear parabolic equations

Computers & Mathematics with Applications, 2016

Gradient schemes are numerical methods, which can be conforming and nonconforming, have been rece... more Gradient schemes are numerical methods, which can be conforming and nonconforming, have been recently developed in Droniou et al. (2013), Droniou et al. (2015), Eymard et al. (2012) and references therein to approximate different types of partial differential equations. They are written in a discrete variational formulation and based on the approximation of functions and gradients. The aim of the present paper is to provide gradient schemes along with an analysis for the convergence order of these schemes for semilinear parabolic equations in any space dimension. We present three gradient schemes. The first two schemes are nonlinear whereas the third one is linear. The existence and uniqueness of the discrete solutions for the first two schemes is proved, thanks to the use of the method of contractive mapping, under the assumption that the mesh size of the time discretization k is small, whereas the existence and uniqueness of the discrete solution for the third scheme is proved for arbitrary k. We provide a convergence rate analysis in discrete semi-norms of L ∞ (H 1) and W 1,2 (L 2) and in the norm of L ∞ (L 2). We prove that the order in space is the same one proved in Eymard et al. (2012) when approximating elliptic equations and one or two in time. The existence, uniqueness, and the convergence results stated above do not require any relation between spacial and temporal discretizations. As an application of these results, we focus on the gradient schemes which use the discrete gradient introduced recently in the SUSHI method (Eymard et al., 2010) and we provide some numerical tests.

Research paper thumbnail of On the discretization of ohmic losses

We consider a nonlinear system of elliptic equations, which arises when modelling the heat diffus... more We consider a nonlinear system of elliptic equations, which arises when modelling the heat diffusion problem coupled with the electrical diffusion problem. The ohmic losses which appear as a source term in the heat diffusion equation yield a nonlinear term which lies in L 1 . A finite element scheme is considered for the discretization of the system; we show that the approximate solution obtained with the scheme converges, up to a subsequence, to a solution of the coupled elliptic system. RÉSUMÉ. On considère ici un système d'équations elliptiques non linéaires, qui provient de la modé-lisation d'un problème de diffusion de la chaleur couplé à un problème de diffusion électrique. L'effet Joule se modélise par un terme non linéaire qui est dans L 1 . On étudie un schéma éléments finis pour la discrétisation de ce système. On montre que la solutions approchée obtenue, converge, à une sous-suite près, vers une solution du problème continu.