Résumé: We apply MFEMs (Mixed Finite Element Methods) to Parabolic Equations. The approach of such MFEM is that it is based on Primal-Dual weak formulation. The discretization in time is performed using Crank-Nicolson method on Uniform temporal meshes. We prove error estimate towards velocity (resp. time derivative of presure) in the divergence norm (resp. L2-norm).

Résumé: We investigate the existence of a weak solution for homogeneous incompressible Bingham fluid is investigated. The rheology of such a fluid is defined by a yield stress and a discontinuous stress–strain law. This non-Newtonian fluid behaves like a solid at low stresses and like a non-linear fluid above the yield stress. In this work we propose to build a weak solution for Navier stokes Bingham equations using a bi-viscosity fluid as an approximation, in particular, we prove that the bi-viscosity tensor converges weakly to the Bingham tensor. This choice allowed us to show the existence of solutions when the source term function is belonging to a convenient space.

Résumé: We discuss the finite volume approximation of elliptic equation with Oblique Boundary Conditions on a part and Dirichlet or Neumann boundary conditions on the other part. An application to an inverse problem is also addressed.

Résumé: We present a fully discrete Mixed Finite Element scheme for Parabolic equations. We prove two new error estimates which corresponding to the convergence towards the time derivative and to the gradient of the exact solution.

Résumé: We consider as discretization in space, the GDM (Gradient Discretization Method) developed recently in Droniou et al. (2018), to approximate multidimensional time fractional diffusion and diffusion-wave equations where the fractional order of the time derivative is respectively satisfying and . The time fractional derivative is given in the Caputo sense. The time discretization is performed using a uniform mesh. For the time fractional diffusion equation, we derive an implicit scheme. In addition to an -a priori estimate, which can be derived from a reasoning in Bradji (2018), we present and prove a new -a priori estimate for the discrete problem. Under the assumption that the exact solution is sufficiently smooth, these a priori estimates allow to prove error estimates in discrete norms of and . The convergence order in these estimates is optimal in the sense of two points of view. The first one is that the order in space is the same one proved in Droniou et al. (2018) for elliptic equation, whereas the second point of view is that the particular case of this order when is the same one obtained in Droniou et al. (2018) for the standard heat equation. For the time fractional diffusion-wave equation, we derive two implicit schemes. A full convergence analysis is carried out for both schemes. In particular, we develop new a priori estimates which yield error estimates in several discrete norms for each scheme. The convergence is proved to be unconditional and optimal. The convergence results are obtained thanks to a comparison with appropriately chosen auxiliary gradient schemes and to the stated a priori estimates. These results improve the ones of Bradji (2017) in which a conditional convergence is proved for a SUSHI (Scheme using Stabilization and Hybrid Interfaces) approximating a time-fractional diffusion-wave equation. For both cases of time fractional diffusion and diffusion-wave equations, we show the well-posedness in the sense that the discrete solutions depend continuously on the data of the considered problems. The stated results can be extended to multi-term time-fractional diffusion and diffusion-wave equations. One of the main features when using GDM is that their results hold for all the schemes within the framework of GDM: conforming and nonconforming finite element, mixed finite element, hybrid mixed mimetic family, some Multi-Point Flux approximation finite volume schemes, and some discontinuous Galerkin schemes. Some examples of schemes recovered by the GSs (Gradient Schemes) presented in this work are sketched. Among these examples, we quote the one presented in Jin et al. (2015) to approximate time fractional diffusion equations. This work extends and improves some results presented in brief or stated without proof in the notes Bradji (2018). We present some numerical tests using SUSHI introduced in Eymard et al. (2010) to support the theoretical results.

Résumé: 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 and to a comparison with an appropriately chosen auxiliary finite volume scheme, we derive the convergence results. This work can be viewed as a continuation of the previous work (Bradji and Gallouët in Int J Finite Vol 3(2):1–35, 2006) where a convergence analysis for a finite volume scheme, based on the admissible mesh of Eymard et al. (In: Ciarlet and Lions, Handbook of numerical analysis, North-Holland, Amsterdam, 2000), for the Poisson’s equation with a linear oblique derivative boundary conditions is given. The obtained convergence results do not require any relation between the mesh sizes of the spatial and time discretizations. Some numerical tests are presented for both elliptic and parabolic equations. In particular, we present three methods to compute the discrete solution.

Résumé: This work is devoted to the convergence analysis of finite volume schemes for a model of semilinear second order hyperbolic equations. The model includes for instance the so-called Sine-Gordon equation which appears for instance in Solid Physics (cf. Fang and Li, Adv Math (China) 42 (2013), 441–457; Liu et al., Numer Methods Partial Differ Equ 31 (2015), 670–690). We are motivated by two works. The first one is Eymard et al. (IMA J Numer Anal 30 (2010), 1009–1043) where a recent class of nonconforming finite volume meshes is introduced. The second one is Eymard et al. (Numer Math 82 (1999), 91–116) where a convergence of a finite volume scheme for semilinear elliptic equations is provided. The mesh considered in Eymard et al. (Numer Math 82 (1999), 91–116) is admissible in the sense of Eymard et al. (Elsevier, Amsterdam, 2000, 723–1020) and a convergence of a family of approximate solutions toward an exact solution when the mesh size tends to zero is proved. This article is also a continuation of our previous two works (Bradji, Numer Methods Partial Differ Equ 29 (2013), 1278–1321; Bradji, Numer Methods Partial Differ Equ 29 (2013), 1–39) which dealt with the convergence analysis of implicit finite volume schemes for the wave equation. We use as discretization in space the generic spatial mesh introduced in Eymard et al. (IMA J Numer Anal 30 (2010), 1009–1043), whereas the discretization in time is performed using a uniform mesh. Two finite volume schemes are derived using the discrete gradient of Eymard et al. (IMA J Numer Anal 30 (2010), 1009–1043). The unknowns of these two schemes are the values at the center of the control volumes, at some internal interfaces, and at the mesh points of the time discretization. The first scheme is inspired from the previous work (Bradji, Numer Methods Partial Differ Equ 29 (2013), 1–39), whereas the second one (in which the discretization in time is performed using a Newmark method) is inspired from the work (Bradji, Numer Methods Partial Differ Equ 29 (2013), 1278–1321). Under the assumption that the mesh size of the time discretization is small, we prove the existence and uniqueness of the discrete solutions. If we assume in addition to this that the exact solution is smooth, we derive and prove three error estimates for each scheme. The first error estimate is concerning an estimate for the error between a discrete gradient of the approximate solution and the gradient of the exact solution whereas the second and the third ones are concerning the estimate for the error between the exact solution and the discrete solution in several discrete seminorm. The existence, uniqueness, and convergence results stated above do not require any relation between k and h.

Résumé: 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 is small, whereas the existence and uniqueness of the discrete solution for the third scheme is proved for arbitrary. We provide a convergence rate analysis in several discrete semi-norms. 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.

Résumé: In this article, we shall present in detail the results announced in a previous work. We consider a model of a linear time dependent Schrödinger equation with a time dependent potential. This model arises, for example, in underwater acoustics and has been studied by Akrivis and Dougalis [3]. We derive a new finite volume scheme on general nonconforming multidimensional spatial meshes introduced recently by Eymard et al. [18] for stationary anisotropic heterogeneous diffusion problems. The unknowns of this scheme are the values at the centre of the control volumes, at some internal interfaces, and at the mesh points of the time discretization. The discretization of the initial condition is performed using a discrete orthogonal projection. A new a priori estimate for the discrete problem is proved. Thanks to this a priori estimate for the discrete problem, we derive error estimates in several discrete norms. Moreover, we establish an error estimate for an approximation for the gradient, in a general framework. We prove that the convergence order is h+k, where h (resp. k) is the mesh size of the spatial (resp. time) discretization. This estimate is valid under the regularity assumption for the exact solution u. These error estimates are useful because they allow us to get error estimates for the approximations of the exact solution and its first derivatives. Results of the present work have been obtained by a comparison with an appropriately chosen auxiliary finite volume scheme.

Résumé: Finite element methods with piecewise polynomial spaces in space for solving the nonstationary heat equation, as a model for parabolic equations are considered. The discretization in time is performed using the Crank-Nicolson method. A new a priori estimate is proved. Thanks to this new a priori estimate, a new error estimate in the discrete norm of $\mathcal{W}^{1,\infty}(\mathcal{L}^2)$ is proved. An $\mathcal{L}^\infty(\mathcal{H}^1)$-error estimate is also shown. These error estimates are useful since they allow us to get second order time accurate approximations for not only the exact solution of the heat equation but also for its first derivatives (both spatial and temporal). Even the proof presented in this note is in some sense standard but the stated $\mathcal{W}^{1,\infty}(\mathcal{L}^2)$-error estimate seems not to be present in the existing literature of the Crank-Nicolson finite element schemes for parabolic equations

Résumé: We consider a family of conforming finite element schemes with piecewise polynomial space of degree $k$ in space for solving the wave equation, as a model for second order hyperbolic equations. The discretization in time is performed using the Newmark method. A new a priori estimate is proved. Thanks to this new a priori estimate, it is proved that the convergence order of the error is $h^k+\tau^2$ in the discrete norms of $\mathcal{L}^{\infty}(0,T;\mathcal{H}^1(\Omega))$ and $\mathcal{W}^{1,\infty}(0,T;\mathcal{L}^2(\Omega))$, where $h$ and $\tau$ are the mesh size of the spatial and temporal discretization, respectively. These error estimates are useful since they allow us to get second order time accurate approximations for not only the exact solution of the wave equation but also for its first derivatives (both spatial and temporal). Even though the proof presented in this note is in some sense standard, the stated error estimates seem not to be present in the existing literature on the finite element methods which use the Newmark method for the wave equation (or general second order hyperbolic equations).

Résumé: The present work is an extension of our previous work (Bradji and Fuhrmann (2011) [4]) which dealt with error analysis of a finite volume scheme of first order (both in time and space) for parabolic equations on general nonconforming multidimensional spatial meshes introduced recently in Eymard et al. (2010) [12]. We aim in this paper to get a higher-order time accurate scheme for a finite volume method for parabolic equations using the same class of spatial generic meshes stated above. We derive a finite volume scheme approximating the heat equation, as a model for parabolic equations, in which the discretization in time is performed using the Crank–Nicolson method. We derive an a priori estimate for the discrete problem and we prove that the error estimate of the finite volume scheme is of order two in time and it is of optimal order in space. The error estimate is analysed in several norms which allow us to derive approximations for the exact solution and its first derivatives (both spatial and temporal) whose the convergence order is two in time and it is optimal in space. We prove in particular, when the discrete flux is calculated using a stabilized discrete gradient, that the convergence order is k2+hDk2+hD, where hDhD (resp. k ) is the mesh size of the spatial (resp. time) discretization. These estimates are valid under the regularity assumption $C^2(C^2)$ for the exact solution u. The proof of these error estimates is based essentially on a new a priori estimate for the discrete problem and a comparison between the finite volume approximate solution and an auxiliary finite volume approximation.

Résumé: We consider the wave equation, on a multidimensional spatial domain. The discretization of the spatial domain is performed using a general class of nonconforming meshes which has been recently studied for stationary anisotropic heterogeneous diffusion problems, see Eymard et al. (IMAJ Numer Anal 30 (2010), 1009–1043). The discretization in time is performed using a uniform mesh. We derive a new implicit finite volume scheme approximating the wave equation and we prove error estimates of the finite volume approximate solution in several norms which allow us to derive error estimates for the approximations of the exact solution and its first derivatives. We prove in particular, when the discrete flux is calculated using a stabilized discrete gradient, the convergence order is equation image (resp. k) is the mesh size of the spatial (resp. time) discretization. This estimate is valid under the regularity assumption equation $C^3(C^2)$ for the exact solution u. The proof of these error estimates is based essentially on a comparison between the finite volume approximate solution and an auxiliary finite volume approximation

Résumé: We present a high order time accurate finite volume scheme for the wave equation.

Résumé: A general class of nonconforming meshes has been recently used to approximate stationary anisotropic heterogeneous diffusion problems in any space dimensions. The aim of the present work is to deal with some error estimates of the discretization of parabolic equations on this general class of meshes in several space dimensions. We present an implicit scheme based on an orthogonal projection of the exact initial function. We provide error estimates in discrete norms View the MathML source $L^\infty(H1$ and $W^{1,\infty}(L2)$.

Résumé: 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 couples the equations. A finite-element scheme and a finite-volume scheme are considered for the discretization of the system; in both cases, we show that the approximate solution obtained with the scheme converges, up to a subsequence, to a solution of the coupled elliptic system.

Résumé: An implicit finite volume scheme for parabolic equations, in which the approximate initial condition is an orthogonal projection of the exact initial function, is considered. In this Note, we prove that the error estimate is of order (where h and k are, respectively, the mesh size of the space discretization and the mesh size of the time discretization) on the discrete norms of $L^\infty(H1)$ and $W^{1,\infty}(L2)$. From these results, error estimates can be derived for the approximations of the fluxes across the interfaces between neighbouring control volumes and of the first derivative of the unknown solution with respect to the time

Résumé: In this paper, we analyse the ‘defect-correction’ technique on a general smooth region, via composite finite-element meshes (a Cartesian mesh and a polar mesh) on two overlapping subdomains (a rectangle and an annulus). Boundary interpolatory mappings of higher degree are used, in the Schwarz method, to pass from one mesh to another. An explicit relation is given between the degree of these mappings and the number of optimal corrections to be computed. Optimal convergence results for the discrete bilinear basic solution, in higher-order discrete Sobolev norms, are obtained on the subdomains. Because the success of the defect-correction technique is based on the uniformity of the discretization and the regularity of the exact solution, the defects are computed on the subdomains in the same way as for the basic solution. Optimal second order improvement per correction is obtained. Numerical results are presented to support the theory.

Résumé: We improve the convergence order of finite volume methods in two dimensions.

Résumé: We improve the convergence order of finite volume methods in one dimension.

Résumé: This paper is an improvement of a previous work, concerning the Laplace equation with an oblique boundary condition. When the boundary condition involves a regular coefficient, we present a weak formulation of the problem and we prove some existence and uniqueness results of the weak solution. We develop a Finite volume scheme and we prove the convergence of the Finite volume solution to the weak solution, when the mesh size goes to zero. We also present some partial results for the interesting case of a discontinuous coefficient in the boundary condition. In particular, we give a Finite volume scheme, taking in consideration the discontinuities of this coefficient Finally, we obtain some error estimates (in a convenient norm) of order h (where h is the mesh size), when the solution u is regular enough.

Résumé: We first establish a finite volume approximating a Distributed Order Diffusion Equation. We subsequently, analyse its convergence.

Résumé: We apply SUSHI method to Bingham Flow Model. The model is a nonlinear problem. We prove the well posedness and the convergence of the scheme.

Résumé: We apply SUSHI to a Time Fractional Diffusion Equation with Delay. The convergence of the scheme is proved.

Résumé: We justify the superconvergence, in the divergence norm, of Low Order Mixed Finite Element methods applied to one dimensional parabolic equations

Résumé: In this work, we extend the results of a previous work to some second order time accurate GSs (Gradient Schemes) applied to a general TFDE (Time Fractional Diffusion Equation) with a space-dependent conductivity. The time fractional derivative is taken in the Caputo sense. The space discretization is performed using the general framework of GDM (Gradient Discretization Method) which encompasses several numerical methods. The approximation of the Caputo derivative is given by the known $L2-1_\sigma$-formula. We prove a new discrete -a priori estimate which, in turn, helps establishing a new -error estimate for the stated second order time accurate GSs. The GDM considered in this work is restricted to the cases of the numerical methods in which the reconstruction operator of the approximate functions satisfies some suitable condition.

Résumé: Some new error estimates for the standard mixed finite element methods for parabolic equations are presented.

Résumé: We provide a finite volume scheme approximation Wave Equation with Several Time Independent Delays and analyse its convergence.

Résumé: Finite volume methods are applied to approximate the wave equations with a constant delay.

Résumé: Finite volume methods are applied to approximate multi-dimensional finite volume scheme for a time-fractional diffusion-wave equation.

Résumé: Finite volume methods are applied to approximate the time fractional heat equation.

Résumé: Some new high order finite element approximations are presented to approximate the wave equation in one space dimension. These high order approximations can be computed using the low order schemes.

Résumé: We consider the piecewise linear finite element method for solving one dimensional second order elliptic equations on a general mesh. It is known that the error between the exact solution and the finite element approximate solution is of first order in the energy norm, whereas the piecewise linear interpolation of the stated error is of second order in the energy norm. In this note, we construct a new piecewise linear approximation such that the piecewise linear interpolation of the error between the exact solution and this new approximation is of fourth order in the energy norm. In addition, the resulting matrix is exactly the same as the one resulting from the finite element approximation while the right hand side is corrected. It is worth mentioning that, by adding some suitable expression to this new fourth order piecewise linear approximation, we obtain a fourth order approximation in $L^2$-norm for the exact solution itself in the energy norm. The present note is an initiation for a work aiming to establish high--order piecewise linear (or bilinear) finite element approximations on general meshes for different types of partial differential equations in several space dimension.

Résumé: We present some new discrete a priori estimates results involving $L^1$-estimate on right hand side for finite volume schemes of linear parabolic equations using admissible meshes. These a priori estimates results are useful since they appear when handling the error in the discretization of a time dependent Joule heating system. The present note is an initiation for a work aiming to provide a convergence analysis of a finite volume scheme of a time dependent Joule heating and it can be viewed as a continuation for Bradji and Herbin (IMAJNA, 2008) where some appropriate schemes are used to approximate a stationary case of a coupled system.

Résumé: This note is a continuation for our work Bradji and Fuhrmann (Applications of Mathematics, Praha, 58/1, 2013 ) which dealt with the convergence analysis of implicit finite volume schemes for linear parabolic equations. This contribution is devoted to the case of semilinear parabolic equations. We use as discretization in space the generic spatial mesh introduced in Eymard et al. (IMAJNA, 2010), whereas the discretization in time is performed using a uniform mesh. We present a finite volume scheme defined using the discrete gradient of Eymard et al. (IMAJNA, 2010). The unknowns of this scheme are the values at the centre of the control volumes, at some internal interfaces, and at the mesh points of the time discretization. Existence, uniqueness of the discrete solution and some error estimates are proved.

Résumé: New finite volume schemes are presented to approximate oblique derivative boundary value problems.

Résumé: A new finite volume scheme along with a convergence analysis are presented for a linear Schrödinger evolution equation.

Résumé: Some new high order finite volume approximations are presented for the wave equation in one space dimension

Résumé: A new second order time accurate finite volume scheme is presented for the wave equation.

Résumé: A new convergence analysis for a finite volume scheme for the wave equation on general nonconforming multidimensional spatial meshes is performed.

Résumé: A new approach is presented to improve the convergence order in finite volume and finite element methods using arbitrary meshes and low order schemes. In particular, the computational cost of these high order approximation is not much.

Résumé: Some convergence results are presented for a model of Ohmic Losses. Such model is a nonlinear coupled problem.

Résumé: New schemes for a coupled heat and electrical diffusion problem are presented.

Résumé: New schemes for a model in Ohmic losses are presented.

Résumé: A convergence of a new finite volume scheme for an oblique derivative boundary problem is performed. The finite volume mesh is admissible.

Résumé: Some new high order approximations, using finite element and finite volume methods, are presented.

Résumé: We justify the superconvergence of Low Order Mixed Finite Element methods in the divergence norm applied to One Dimensional Parabolic Equations.

Résumé: We present an error estimate, in energy norm, for Gradient Discretization Method combined with a Crank-Nicolson type method applied to time fractional diffusion equations.

Résumé: We presented some new results obtained in the context of Mixed Finite Methods.

Résumé: We establish a finite volume scheme for a simple Bingham Flow Model and analyse its convergence.

Résumé: We consider a finite volume scheme for a Distributed Order Diffusion Equation and provide its error estimates.

Résumé: We construct a finite volume scheme for a Time Fractional Diffusion Equation with Delay and study its convergence.

Résumé: We present new error estimates for Mixed Finite Element Methods applied to Parabolic equations. These error estimates are given in the energy norm for velocity.

Résumé: In this talk, we first provide some existing results in Non-Newtonian Fluids and then we give some new results of existence for a Bingham Flow. This talk was presented by Aboussi in presence.

Résumé: We establish a Finite Volume scheme for a Distributed Fractional Derivative. Distributed Fractional Derivative is defined over an weighted integral over (0,1) to which the fractional derivative is belonging. This talk was presented by me online.

Résumé: A finite volume scheme is presented for a wave equation with several time independent delays along a convergence analysis.

Résumé: We present a new error estimate for a Primal-Dual Crank Nicolson Mixed Finite Element Using Lowest Degree Raviart Thomas Spaces for Parabolic Equations.

Résumé: Some new discrete a priori estimates for a finite volume scheme appearing in the discretization of a time dependent Joule heating system are proved. Such estimates allow to prove the convergence of the family of finite volume approximations towards an exact solution of a weak formulation.

Résumé: We present some results concerning a numerical scheme for model of ohmic losses.

Résumé: We present a numerical scheme for a model of Ohmic Losses

Résumé: 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 volume scheme is proposed 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: We present a new method to improve the convergence of finite volume methods.

Résumé: A defect correction method is presented for a second order elliptic equation posed on a disc in two dimensions.