Mejdi Azaiez - Profile on Academia.edu (original) (raw)

Papers by Mejdi Azaiez

Research paper thumbnail of A new hp method for the operator in non-Cartesian geometries

Applied Numerical Mathematics, 2008

This paper describes a new hp method for the numerical solution of the two-dimensional −grad(div)... more This paper describes a new hp method for the numerical solution of the two-dimensional −grad(div) eigenvalue problem in primal variational formulation. In standard methods with triangular or quadrangular finite elements, or with spectral elements, the spectrum contains spurious modes if the mesh is non-Cartesian. With the new hp method described in this paper we show that, although it delivers a P p-P p approximation on the same grid for both velocity components, the spectrum is represented without any spurious mode whether a Cartesian or a non-Cartesian quadrangular grid is chosen. Spectra computed with standard high-order methods and with the new element are presented and compared with the analytic solutions.

Research paper thumbnail of Convergence de la POD pour des problèmes multiparamètre

On s'intérsse ‡ l'analyse numérique de la convergence de la POD pour représenter des solutions de... more On s'intérsse ‡ l'analyse numérique de la convergence de la POD pour représenter des solutions de l'équation de la chaleur paramétisée. La paramètre étant la condictivité. A l'aide de plusiers exemples numériques nous montrons la convergence exponentielle en fonction des modes POD. Ensuite nous apportons quelques éléments pour la justification mathématique de ce résultat.

Research paper thumbnail of An efficient spectral element solver of Poisson problem

An efficient spectral element solver of Poisson problem

Comptes Rendus de l Académie des Sciences - Series I - Mathematics

Research paper thumbnail of Optimal spectral direct solver of the operator (αI+curlcurl)

Optimal spectral direct solver of the operator (αI+curlcurl)

Comptes Rendus de l Académie des Sciences - Series I - Mathematics

Research paper thumbnail of Spectral methods applied to porous media equations

Spectral methods applied to porous media equations

East-West Journal of Numerical Mathematics

ABSTRACT

Research paper thumbnail of Dual spectral element methods for second-order axisymmetric problems. Application to the Goda projection algorithm. (Méthodes d’éléments spectreaux pour des problèmes hybrides duaux de second ordre axisymétriques. Application à l’algorithme de projection de Goda.)

Dual spectral element methods for second-order axisymmetric problems. Application to the Goda projection algorithm. (Méthodes d’éléments spectreaux pour des problèmes hybrides duaux de second ordre axisymétriques. Application à l’algorithme de projection de Goda.)

Revue européenne des éléments finis

Research paper thumbnail of The density function reconstruction of surface sources from a single Cauchy measurement

Computers & Fluids, 2011

The inverse problem of reconstructing sources is explored when a single boundary Cauchy data is p... more The inverse problem of reconstructing sources is explored when a single boundary Cauchy data is postulated on the potential. We are particularly involved in sources supported by (hyper-)surfaces. Mild assumptions are required on the location of these supports and the calculation of the charge density function is then aimed. We consider a variational formulation, based on a duplication artifice of the potential and we check the symmetry and the positive definiteness of the weak problem. Because of the severe illposedness, the use of a regularization is mandatory for a safe approximation of the solution. Lavrentiev's method is therefore recommended in the context owing to the symmetry and the positivity. We check why that regularization turns out to be a Tikhonov method for some underlying shadow equation that is not needed in computations and is therefore never explicitly constructed. Results stated in a wide literature for the Tikhonov regularization applies as well to our variational problem. An important consequence is that the Morozov Discrepancy Principle, we use for the selection of the regularization parameter yields a convergent strategy. Now, that the Discrepancy Principle requires the residual of that inaccessible 'shadow equation', we explain how the Kohn-Vogelius function allows for the computation of that residual.

Research paper thumbnail of Numerical assessment of a class of uniformly stable mixed spectral elements for the Navier–Stokes equations

Computers & Fluids, 2007

In 1999, Bernardi and Maday analyzed a new class of mixed spectral elements for the Stokes and th... more In 1999, Bernardi and Maday analyzed a new class of mixed spectral elements for the Stokes and the Navier-Stokes equations [Bernardi C, Maday Y. Uniform Inf-Sup condition for the spectral discretization of the Stokes problem. Math Models Meth Appl Sci 1999;3:395-414] where they proved some interesting results like the uniform Inf-Sup condition. The main advantage we see is that applying the Uzawa algorithm to the discrete Stokes system yields a well-conditioned problem on the pressure. Then, the mass matrix preconditioned Conjugate Gradient method PCG used to compute the pressure converges in a number of iterations that is independent of the polynomial degree approximation. This paper presents the ''numerical proofs'' of the theoretical predictions on the stability and the accuracy of these spectral methods in mono-domain and multi-domain configurations.

Research paper thumbnail of Onset of a Double-Diffusive Convective Regime in a Rectangular Porous Cavity

Onset of a Double-Diffusive Convective Regime in a Rectangular Porous Cavity

Journal of Porous Media, 1998

... Marie-Catherine Charrier-Mojtabi Laboratoire PHASE EA 3028, Université Paul Sabatier, France.... more ... Marie-Catherine Charrier-Mojtabi Laboratoire PHASE EA 3028, Université Paul Sabatier, France.Mohammad Karimi-Fard Institut de Mécanique des Fluides, UMR CNRS-INP-UPS №5502, Avenue du Professeur Camille Soula, F 31400 Toulouse, France. ...

Research paper thumbnail of Double-diffusive convection in an annular verticalporous layer

Double-diffusive convection in an annular verticalporous layer

International Journal of Heat and Mass Transfer, 1999

This paper reports an analytical and numerical study of double-diffusive naturalconvection throug... more This paper reports an analytical and numerical study of double-diffusive naturalconvection through a fluid-saturated, vertical and homogeneous porous annulus subjected touniform fluxes of heat and mass from the side. The influence of each leading parameter ...

Research paper thumbnail of Numerical and experimental study of multicellular free convection flows in an annular porous layer

Numerical and experimental study of multicellular free convection flows in an annular porous layer

International Journal of Heat and Mass Transfer, 1991

... C. Canute, MY Hussaini, TA Zang and A. Quar-sional natural convection in a porous medium betw... more ... C. Canute, MY Hussaini, TA Zang and A. Quar-sional natural convection in a porous medium between teroni, Spectral Methods in ... Diese Strukturen wurden bisher noch nie in konzentrischen Ringr umen mit Hilfe des Christiansen-Effekts beobachtet, sie stimmen gut mit den ...

Research paper thumbnail of A unique grid spectral solver of the nD Cartesian unsteady Strokes system. Illustrative numerical results

A unique grid spectral solver of the nD Cartesian unsteady Strokes system. Illustrative numerical results

Finite Elements in Analysis and Design, 1994

ABSTRACT

Research paper thumbnail of Streamline derivative projection-based POD-ROM for convection-dominated flows. Part I : Numerical Analysis

arXiv: Numerical Analysis, 2017

We introduce improved Reduced Order Models (ROM) for convection-dominated flows. These non-linear... more We introduce improved Reduced Order Models (ROM) for convection-dominated flows. These non-linear closure models are inspired from successful numerical stabilization techniques used in Large Eddy Simulations (LES), such as Local Projection Stabilization (LPS), applied to standard models created by Proper Orthogonal Decomposition (POD) of flows with Galerkin projection. The numerical analysis of the fully Navier-Stokes discretization for the proposed new POD-ROM is presented, by mainly deriving the corresponding error estimates. Also, we suggest an efficient practical implementation of the stabilization term, where the stabilization parameter is approximated by the Discrete Empirical Interpolation Method (DEIM).

Research paper thumbnail of A cure for instabilities due to advection-dominance in POD solution to advection-diffusion-reaction equations

Journal of Computational Physics, 2021

In this paper, we propose to improve the stabilized POD-ROM introduced by S. Rubino in [37] to de... more In this paper, we propose to improve the stabilized POD-ROM introduced by S. Rubino in [37] to deal with the numerical simulation of advection-dominated advectiondiffusion-reaction equations. In particular, we introduce a stabilizing post-processing strategy that will be very useful when considering very low diffusion coefficients, i.e. in the strongly advection-dominated regime. This strategy is applied both for the offline phase, to produce the snapshots, and the reduced order method to simulate the new solutions. The new process of a posteriori stabilization is detailed in a general framework and applied to advection-diffusion-reaction problems. Numerical studies are performed to discuss the accuracy and performance of the new method in handling strongly advection-dominated cases.

Research paper thumbnail of New Unconditionally Stable Schemes for the Navier-Stokes Equations

Communications in Computational Physics, 2021

In this paper we propose some efficient schemes for the Navier-Stokes equations. The proposed sch... more In this paper we propose some efficient schemes for the Navier-Stokes equations. The proposed schemes are constructed based on an auxiliary variable reformulation of the underlying equations, recently introduced by Li et al. [20]. Our objective is to construct and analyze improved schemes, which overcome some of the shortcomings of the existing schemes. In particular, our new schemes have the capability to capture steady solutions for large Reynolds numbers and time step sizes, while keeping the error analysis available. The novelty of our method is twofold: i) Use the Uzawa algorithm to decouple the pressure and the velocity. This is to replace the pressurecorrection method considered in [20]. ii) Inspired by the paper [21], we modify the algorithm using an ingredient to capture stationary solutions. In all cases we analyze a first-and second-order schemes and prove the unconditionally energy stability. We also provide an error analysis for the first-order scheme. Finally we validate our schemes by performing simulations of the Kovasznay flow and double lid driven cavity flow. These flow simulations at high Reynolds numbers demonstrate the robustness and efficiency of the proposed schemes.

Research paper thumbnail of An unconditionally stable fast high order method for thermal phase change models

An unconditionally stable fast high order method for thermal phase change models

Computers & Fluids, 2022

Research paper thumbnail of New efficient time-stepping schemes for the anisotropic phase-field dendritic crystal growth model

Computers & Mathematics with Applications, 2022

In this paper, we propose and analyze a first-order and a second-order time-stepping schemes for ... more In this paper, we propose and analyze a first-order and a second-order time-stepping schemes for the anisotropic phase-field dendritic crystal growth model. The proposed schemes are based on an auxiliary variable approach for the Allen-Cahn equation and delicate treatment of the terms coupling the Allen-Cahn equation and temperature equation. The idea of the former is to introduce suitable auxiliary variables to facilitate construction of high order stable schemes for a large class of gradient flows. We propose a new technique to treat the coupling terms involved in the crystal growth model, and introduce suitable stabilization terms to result in totally decoupled schemes, which satisfy a discrete energy law without affecting the convergence order. A delicate implementation demonstrates that the proposed schemes can be realized in a very efficient way. That is, it only requires solving four linear elliptic equations and a simple algebraic equation at each time step. A detailed comparison with existing schemes is given, and the advantage of the new schemes are emphasized. As far as we know this is the first second-order scheme that is totally decoupled, linear, unconditionally stable for the dendritic crystal growth model with variable mobility parameter.

Research paper thumbnail of An intrinsic Proper Generalized Decomposition for parametric symmetric elliptic problems

We introduce in this paper a technique for the reduced order approximation of parametric symmetri... more We introduce in this paper a technique for the reduced order approximation of parametric symmetric elliptic partial differential equations. For any given dimension, we prove the existence of an optimal subspace of at most that dimension which realizes the best approximation in mean of the error with respect to the parameter in the quadratic norm associated to the elliptic operator, between the exact solution and the Galerkin solution calculated on the subspace. This is analogous to the best approximation property of the Proper Orthogonal Decomposition (POD) subspaces, excepting that in our case the norm is parameter-depending, and then the POD optimal sub-spaces cannot be characterized by means of a spectral problem. We apply a deflation technique to build a series of approximating solutions on finite-dimensional optimal subspaces, directly in the on-line step. We prove that the partial sums converge to the continuous solutions, in mean quadratic elliptic norm.

Research paper thumbnail of An extension of the landweber regularization for a backward time fractional wave problem

Discrete & Continuous Dynamical Systems - S, 2018

In this paper, we investigate numerical methods for a backward problem of the time-fractional wav... more In this paper, we investigate numerical methods for a backward problem of the time-fractional wave equation in bounded domains. We propose two fractional filter regularization methods, which can be regarded as an extension of the classical Landweber iteration for the time-fractional wave backward problem. The idea is first to transform the ill-posed backward problem into a weighted normal operator equation, then construct the regularization methods for the operator equation by introducing suitable fractional filters. Both a priori and a posteriori regularization parameter choice rules are investigated, together with an estimate for the smallest regularization parameter according to a discrepancy principle. Furthermore, an error analysis is carried out to derive the convergence rates of the regularized solutions generated by the proposed methods. The theoretical estimate shows that the proposed fractional regularizations efficiently overcome the well-known over-smoothing drawback caused by the classical regularizations. Some numerical examples are provided to confirm the theoretical results. In particular, our numerical tests demonstrate that the fractional regularization is actually more efficient than the classical methods for problems having low regularity.

Research paper thumbnail of A Müntz-Collocation Spectral Method for Weakly Singular Volterra Integral Equations

Journal of Scientific Computing, 2019

In this paper we propose and analyze a fractional Jacobi-collocation spectral method for the seco... more In this paper we propose and analyze a fractional Jacobi-collocation spectral method for the second kind Volterra integral equations (VIEs) with weakly singular kernel (x − s) −µ , 0 < µ < 1. First we develop a family of fractional Jacobi polynomials, along with basic approximation results for some weighted projection and interpolation operators defined in suitable weighted Sobolev spaces. Then we construct an efficient fractional Jacobi-collocation spectral method for the VIEs using the zeros of the new developed fractional Jacobi polynomial. A detailed convergence analysis is carried out to derive error estimates of the numerical solution in both L ∞-and weighted L 2-norms. The main novelty of the paper is that the proposed method is highly efficient for typical solutions that VIEs usually possess. Precisely, it is proved that the exponential convergence rate can be achieved for solutions which are smooth after the variable change x → x 1/λ for a suitable real number λ. Finally a series of numerical examples are presented to demonstrate the efficiency of the method.

Research paper thumbnail of A new hp method for the operator in non-Cartesian geometries

Applied Numerical Mathematics, 2008

This paper describes a new hp method for the numerical solution of the two-dimensional −grad(div)... more This paper describes a new hp method for the numerical solution of the two-dimensional −grad(div) eigenvalue problem in primal variational formulation. In standard methods with triangular or quadrangular finite elements, or with spectral elements, the spectrum contains spurious modes if the mesh is non-Cartesian. With the new hp method described in this paper we show that, although it delivers a P p-P p approximation on the same grid for both velocity components, the spectrum is represented without any spurious mode whether a Cartesian or a non-Cartesian quadrangular grid is chosen. Spectra computed with standard high-order methods and with the new element are presented and compared with the analytic solutions.

Research paper thumbnail of Convergence de la POD pour des problèmes multiparamètre

On s'intérsse ‡ l'analyse numérique de la convergence de la POD pour représenter des solutions de... more On s'intérsse ‡ l'analyse numérique de la convergence de la POD pour représenter des solutions de l'équation de la chaleur paramétisée. La paramètre étant la condictivité. A l'aide de plusiers exemples numériques nous montrons la convergence exponentielle en fonction des modes POD. Ensuite nous apportons quelques éléments pour la justification mathématique de ce résultat.

Research paper thumbnail of An efficient spectral element solver of Poisson problem

An efficient spectral element solver of Poisson problem

Comptes Rendus de l Académie des Sciences - Series I - Mathematics

Research paper thumbnail of Optimal spectral direct solver of the operator (αI+curlcurl)

Optimal spectral direct solver of the operator (αI+curlcurl)

Comptes Rendus de l Académie des Sciences - Series I - Mathematics

Research paper thumbnail of Spectral methods applied to porous media equations

Spectral methods applied to porous media equations

East-West Journal of Numerical Mathematics

ABSTRACT

Research paper thumbnail of Dual spectral element methods for second-order axisymmetric problems. Application to the Goda projection algorithm. (Méthodes d’éléments spectreaux pour des problèmes hybrides duaux de second ordre axisymétriques. Application à l’algorithme de projection de Goda.)

Dual spectral element methods for second-order axisymmetric problems. Application to the Goda projection algorithm. (Méthodes d’éléments spectreaux pour des problèmes hybrides duaux de second ordre axisymétriques. Application à l’algorithme de projection de Goda.)

Revue européenne des éléments finis

Research paper thumbnail of The density function reconstruction of surface sources from a single Cauchy measurement

Computers & Fluids, 2011

The inverse problem of reconstructing sources is explored when a single boundary Cauchy data is p... more The inverse problem of reconstructing sources is explored when a single boundary Cauchy data is postulated on the potential. We are particularly involved in sources supported by (hyper-)surfaces. Mild assumptions are required on the location of these supports and the calculation of the charge density function is then aimed. We consider a variational formulation, based on a duplication artifice of the potential and we check the symmetry and the positive definiteness of the weak problem. Because of the severe illposedness, the use of a regularization is mandatory for a safe approximation of the solution. Lavrentiev's method is therefore recommended in the context owing to the symmetry and the positivity. We check why that regularization turns out to be a Tikhonov method for some underlying shadow equation that is not needed in computations and is therefore never explicitly constructed. Results stated in a wide literature for the Tikhonov regularization applies as well to our variational problem. An important consequence is that the Morozov Discrepancy Principle, we use for the selection of the regularization parameter yields a convergent strategy. Now, that the Discrepancy Principle requires the residual of that inaccessible 'shadow equation', we explain how the Kohn-Vogelius function allows for the computation of that residual.

Research paper thumbnail of Numerical assessment of a class of uniformly stable mixed spectral elements for the Navier–Stokes equations

Computers & Fluids, 2007

In 1999, Bernardi and Maday analyzed a new class of mixed spectral elements for the Stokes and th... more In 1999, Bernardi and Maday analyzed a new class of mixed spectral elements for the Stokes and the Navier-Stokes equations [Bernardi C, Maday Y. Uniform Inf-Sup condition for the spectral discretization of the Stokes problem. Math Models Meth Appl Sci 1999;3:395-414] where they proved some interesting results like the uniform Inf-Sup condition. The main advantage we see is that applying the Uzawa algorithm to the discrete Stokes system yields a well-conditioned problem on the pressure. Then, the mass matrix preconditioned Conjugate Gradient method PCG used to compute the pressure converges in a number of iterations that is independent of the polynomial degree approximation. This paper presents the ''numerical proofs'' of the theoretical predictions on the stability and the accuracy of these spectral methods in mono-domain and multi-domain configurations.

Research paper thumbnail of Onset of a Double-Diffusive Convective Regime in a Rectangular Porous Cavity

Onset of a Double-Diffusive Convective Regime in a Rectangular Porous Cavity

Journal of Porous Media, 1998

... Marie-Catherine Charrier-Mojtabi Laboratoire PHASE EA 3028, Université Paul Sabatier, France.... more ... Marie-Catherine Charrier-Mojtabi Laboratoire PHASE EA 3028, Université Paul Sabatier, France.Mohammad Karimi-Fard Institut de Mécanique des Fluides, UMR CNRS-INP-UPS №5502, Avenue du Professeur Camille Soula, F 31400 Toulouse, France. ...

Research paper thumbnail of Double-diffusive convection in an annular verticalporous layer

Double-diffusive convection in an annular verticalporous layer

International Journal of Heat and Mass Transfer, 1999

This paper reports an analytical and numerical study of double-diffusive naturalconvection throug... more This paper reports an analytical and numerical study of double-diffusive naturalconvection through a fluid-saturated, vertical and homogeneous porous annulus subjected touniform fluxes of heat and mass from the side. The influence of each leading parameter ...

Research paper thumbnail of Numerical and experimental study of multicellular free convection flows in an annular porous layer

Numerical and experimental study of multicellular free convection flows in an annular porous layer

International Journal of Heat and Mass Transfer, 1991

... C. Canute, MY Hussaini, TA Zang and A. Quar-sional natural convection in a porous medium betw... more ... C. Canute, MY Hussaini, TA Zang and A. Quar-sional natural convection in a porous medium between teroni, Spectral Methods in ... Diese Strukturen wurden bisher noch nie in konzentrischen Ringr umen mit Hilfe des Christiansen-Effekts beobachtet, sie stimmen gut mit den ...

Research paper thumbnail of A unique grid spectral solver of the nD Cartesian unsteady Strokes system. Illustrative numerical results

A unique grid spectral solver of the nD Cartesian unsteady Strokes system. Illustrative numerical results

Finite Elements in Analysis and Design, 1994

ABSTRACT

Research paper thumbnail of Streamline derivative projection-based POD-ROM for convection-dominated flows. Part I : Numerical Analysis

arXiv: Numerical Analysis, 2017

We introduce improved Reduced Order Models (ROM) for convection-dominated flows. These non-linear... more We introduce improved Reduced Order Models (ROM) for convection-dominated flows. These non-linear closure models are inspired from successful numerical stabilization techniques used in Large Eddy Simulations (LES), such as Local Projection Stabilization (LPS), applied to standard models created by Proper Orthogonal Decomposition (POD) of flows with Galerkin projection. The numerical analysis of the fully Navier-Stokes discretization for the proposed new POD-ROM is presented, by mainly deriving the corresponding error estimates. Also, we suggest an efficient practical implementation of the stabilization term, where the stabilization parameter is approximated by the Discrete Empirical Interpolation Method (DEIM).

Research paper thumbnail of A cure for instabilities due to advection-dominance in POD solution to advection-diffusion-reaction equations

Journal of Computational Physics, 2021

In this paper, we propose to improve the stabilized POD-ROM introduced by S. Rubino in [37] to de... more In this paper, we propose to improve the stabilized POD-ROM introduced by S. Rubino in [37] to deal with the numerical simulation of advection-dominated advectiondiffusion-reaction equations. In particular, we introduce a stabilizing post-processing strategy that will be very useful when considering very low diffusion coefficients, i.e. in the strongly advection-dominated regime. This strategy is applied both for the offline phase, to produce the snapshots, and the reduced order method to simulate the new solutions. The new process of a posteriori stabilization is detailed in a general framework and applied to advection-diffusion-reaction problems. Numerical studies are performed to discuss the accuracy and performance of the new method in handling strongly advection-dominated cases.

Research paper thumbnail of New Unconditionally Stable Schemes for the Navier-Stokes Equations

Communications in Computational Physics, 2021

In this paper we propose some efficient schemes for the Navier-Stokes equations. The proposed sch... more In this paper we propose some efficient schemes for the Navier-Stokes equations. The proposed schemes are constructed based on an auxiliary variable reformulation of the underlying equations, recently introduced by Li et al. [20]. Our objective is to construct and analyze improved schemes, which overcome some of the shortcomings of the existing schemes. In particular, our new schemes have the capability to capture steady solutions for large Reynolds numbers and time step sizes, while keeping the error analysis available. The novelty of our method is twofold: i) Use the Uzawa algorithm to decouple the pressure and the velocity. This is to replace the pressurecorrection method considered in [20]. ii) Inspired by the paper [21], we modify the algorithm using an ingredient to capture stationary solutions. In all cases we analyze a first-and second-order schemes and prove the unconditionally energy stability. We also provide an error analysis for the first-order scheme. Finally we validate our schemes by performing simulations of the Kovasznay flow and double lid driven cavity flow. These flow simulations at high Reynolds numbers demonstrate the robustness and efficiency of the proposed schemes.

Research paper thumbnail of An unconditionally stable fast high order method for thermal phase change models

An unconditionally stable fast high order method for thermal phase change models

Computers & Fluids, 2022

Research paper thumbnail of New efficient time-stepping schemes for the anisotropic phase-field dendritic crystal growth model

Computers & Mathematics with Applications, 2022

In this paper, we propose and analyze a first-order and a second-order time-stepping schemes for ... more In this paper, we propose and analyze a first-order and a second-order time-stepping schemes for the anisotropic phase-field dendritic crystal growth model. The proposed schemes are based on an auxiliary variable approach for the Allen-Cahn equation and delicate treatment of the terms coupling the Allen-Cahn equation and temperature equation. The idea of the former is to introduce suitable auxiliary variables to facilitate construction of high order stable schemes for a large class of gradient flows. We propose a new technique to treat the coupling terms involved in the crystal growth model, and introduce suitable stabilization terms to result in totally decoupled schemes, which satisfy a discrete energy law without affecting the convergence order. A delicate implementation demonstrates that the proposed schemes can be realized in a very efficient way. That is, it only requires solving four linear elliptic equations and a simple algebraic equation at each time step. A detailed comparison with existing schemes is given, and the advantage of the new schemes are emphasized. As far as we know this is the first second-order scheme that is totally decoupled, linear, unconditionally stable for the dendritic crystal growth model with variable mobility parameter.

Research paper thumbnail of An intrinsic Proper Generalized Decomposition for parametric symmetric elliptic problems

We introduce in this paper a technique for the reduced order approximation of parametric symmetri... more We introduce in this paper a technique for the reduced order approximation of parametric symmetric elliptic partial differential equations. For any given dimension, we prove the existence of an optimal subspace of at most that dimension which realizes the best approximation in mean of the error with respect to the parameter in the quadratic norm associated to the elliptic operator, between the exact solution and the Galerkin solution calculated on the subspace. This is analogous to the best approximation property of the Proper Orthogonal Decomposition (POD) subspaces, excepting that in our case the norm is parameter-depending, and then the POD optimal sub-spaces cannot be characterized by means of a spectral problem. We apply a deflation technique to build a series of approximating solutions on finite-dimensional optimal subspaces, directly in the on-line step. We prove that the partial sums converge to the continuous solutions, in mean quadratic elliptic norm.

Research paper thumbnail of An extension of the landweber regularization for a backward time fractional wave problem

Discrete & Continuous Dynamical Systems - S, 2018

In this paper, we investigate numerical methods for a backward problem of the time-fractional wav... more In this paper, we investigate numerical methods for a backward problem of the time-fractional wave equation in bounded domains. We propose two fractional filter regularization methods, which can be regarded as an extension of the classical Landweber iteration for the time-fractional wave backward problem. The idea is first to transform the ill-posed backward problem into a weighted normal operator equation, then construct the regularization methods for the operator equation by introducing suitable fractional filters. Both a priori and a posteriori regularization parameter choice rules are investigated, together with an estimate for the smallest regularization parameter according to a discrepancy principle. Furthermore, an error analysis is carried out to derive the convergence rates of the regularized solutions generated by the proposed methods. The theoretical estimate shows that the proposed fractional regularizations efficiently overcome the well-known over-smoothing drawback caused by the classical regularizations. Some numerical examples are provided to confirm the theoretical results. In particular, our numerical tests demonstrate that the fractional regularization is actually more efficient than the classical methods for problems having low regularity.

Research paper thumbnail of A Müntz-Collocation Spectral Method for Weakly Singular Volterra Integral Equations

Journal of Scientific Computing, 2019

In this paper we propose and analyze a fractional Jacobi-collocation spectral method for the seco... more In this paper we propose and analyze a fractional Jacobi-collocation spectral method for the second kind Volterra integral equations (VIEs) with weakly singular kernel (x − s) −µ , 0 < µ < 1. First we develop a family of fractional Jacobi polynomials, along with basic approximation results for some weighted projection and interpolation operators defined in suitable weighted Sobolev spaces. Then we construct an efficient fractional Jacobi-collocation spectral method for the VIEs using the zeros of the new developed fractional Jacobi polynomial. A detailed convergence analysis is carried out to derive error estimates of the numerical solution in both L ∞-and weighted L 2-norms. The main novelty of the paper is that the proposed method is highly efficient for typical solutions that VIEs usually possess. Precisely, it is proved that the exponential convergence rate can be achieved for solutions which are smooth after the variable change x → x 1/λ for a suitable real number λ. Finally a series of numerical examples are presented to demonstrate the efficiency of the method.