arXiv subjects

Amiya K. Pani

Publications and source records attributed to Amiya K. Pani.

36 records · Page 2Linked to original sources

arXiv2021

Backward Euler method for the equations of motion arising in Oldroyd model of order one with nonsmooth initial data

In this paper, a backward Euler method combined with finite element discretization in spatial direction is discussed for the equations of motion arising in the $2D$ Oldroyd model of viscoelastic fluids of order one with the forcing term independent of time or in $L^{\infty}$ in time. It is shown that the estimates of the discrete solution in Dirichlet norm is bounded uniformly in time. Optimal {\it a priori} error estimate in $\textbf{L}^2$-norm is derived for the discrete problem with non-smooth initial data. This estimate is shown to be uniform in time, under the assumption of uniqueness condition. Finally, we present some numerical results to validate our theoretical results.

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arXiv2021

Numerical analysis of a new formulation for the Oseen equations in terms of vorticity and Bernoulli pressure

A variational formulation is introduced for the Oseen equations written in terms of vor\-ti\-city and Bernoulli pressure. The velocity is fully decoupled using the momentum balance equation, and it is later recovered by a post-process. A finite element method is also proposed, consisting in equal-order Nédélec finite elements and piecewise continuous polynomials for the vorticity and the Bernoulli pressure, respectively. The {\it a priori} error analysis is carried out in the $\mathrm{L}^2$-norm for vorticity, pressure, and velocity; under a smallness assumption either on the convecting velocity, or on the mesh parameter. Furthermore, an {\it a posteriori} error estimator is designed and its robustness and efficiency are studied using weighted norms. Finally, a set of numerical examples in 2D and 3D is given, where the error indicator serves to guide adaptive mesh refinement. These tests illustrate the behaviour of the new formulation in typical flow conditions, and they also confirm the theoretical findings.

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arXiv2020

A mixed finite element scheme for biharmonic equation with variable coefficient and von Kármán equations

In this paper, a new mixed finite element scheme using element-wise stabilization is introduced for the biharmonic equation with variable coefficient on Lipschitz polyhedral domains. The proposed scheme doesn't involve any integration along mesh interfaces. The gradient of the solution is approximated by $H({\rm div})$-conforming $BDM_{k+1}$ element or vector valued Lagrange element with order $k+1$, while the solution is approximated by Lagrange element with order $k+2$ for any $k\geq 0$.This scheme can be easily implemented and produces positive definite linear system. We provide a new discrete $H^{2}$-norm stability, which is useful not only in analysis of this scheme but also in $C^{0}$ interior penalty methods and DG methods. Optimal convergences in both discrete $H^{2}$-norm and $L^{2}$-norm are derived. This scheme with its analysis is further generalized to the von Kármán equations. Finally, numerical results verifying the theoretical estimates of the proposed algorithms are also presented.

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arXiv2018

Stabilization of Kelvin-Voigt viscoelastic Fuid Fow model

In this article, stabilization result for the viscoelastic fluid flow problem governed by Kelvin-Voigt model, that is, convergence of the unsteady solution to a steady state solution is proved under the assumption that linearized self-adjoint steady state eigenvalue problem has a minimal positive eigenvalue. Both power and exponential convergence results are derived under various conditions on the forcing function. It is shown that results are valid uniformly in the time relaxation or some times called regularization parameter $κ$ as $κ\to 0$, which in turn, establishes results for the Navier-Stokes system.

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arXiv2018

Optimal error analysis of a FEM for fractional diffusion problems by energy arguments

In this article, the piecewise-linear finite element method (FEM) is applied to approximate the solution of time-fractional diffusion equations on bounded convex domains. Standard energy arguments do not provide satisfactory results for such a problem due to the low regularity of its exact solution. Using a delicate energy analysis, {\it a priori} optimal error bounds in $L^2(Ω)$-, $H^1(Ω)$-norms, and a quasi-optimal bound in $L^{\infty}(Ω)$-norm are derived for the semidiscrete FEM for cases with smooth and nonsmooth initial data. The main tool of our analysis is based on a repeated use of an integral operator and use of a $t^m$ type of weights to take care of the singular behavior of the continuous solution at $t=0.$ The generalized Leibniz formula for fractional derivatives is found to play a key role in our analysis. Numerical experiments are presented to illustrate some of the theoretical results.

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arXiv2017

Error analysis of a finite volume element method for fractional order evolution equations with nonsmooth initial data

In this paper, a finite volume element (FVE) method is considered for spatial approximations of time-fractional diffusion equations involving a Riemann-Liouville fractional derivative of order $α\in (0,1)$ in time. Improving upon earlier results (Karaa {\it et al.}, IMA J. Numer. Anal. 2016), optimal error estimates in $L^2(Ω)$- and $H^1(Ω)$-norms for the semidiscrete problem with smooth and middly smooth initial data, i.e., $v\in H^2(Ω)\cap H^1_0(Ω)$ and $v\in H^1_0(Ω)$ are established. For nonsmooth data, that is, $v\in L^2(Ω)$, the optimal $L^2(Ω)$-error estimate is shown to hold only under an additional assumption on the triangulation, which is known to be satisfied for symmetric triangulations. Superconvergence result is also proved and as a consequence, a quasi-optimal error estimate is established in the $L^\infty(Ω)$-norm. Further, two fully discrete schemes using convolution quadrature in time generated by the backward Euler and the second-order backward difference methods are analyzed, and error estimates are derived for both smooth and nonsmooth initial data. Based on a comparison of the standard Galerkin finite element solution with the FVE solution and exploiting tools for Laplace transforms with semigroup type properties of the FVE solution operator, our analysis is then extended in a unified manner to several time-fractional order evolution problems. Finally, several numerical experiments are conducted to confirm our theoretical findings.

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arXiv2017

On an a posteriori error analysis of mixed finite element Galerkin approximations to a second order wave equation

In this article, a posteriori error analysis is developed for mixed finite element Galerkin approximations to a second order linear hyperbolic equation. Based on mixed elliptic reconstructions and an integration tool, which is a variation of Baker's technique introduced earlier by G. Baker ( SIAM J. Numer. Anal., 13 (1976), 564-576) in the context of a priori estimates for a second order wave equation, a posteriori error estimates of the displacement in L{\infty}(L2)-norm for the semidiscrete scheme are derived under minimal regularity. Finally, a first order implicit-in-time discrete scheme is analyzed and a posteriori error estimators are established.

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arXiv2016

Approximate Controllability of a Class of Partial Integro-Differential Equations of Parabolic Type

In this paper, we discuss the distributed control problem governed by the following parabolic integro-differential equation (PIDE) in the abstract form \begin{eqnarray*} \frac{\partial y}{\partial t} + A y &=& \int_0^t B(t, s) y(s) ds + Gu, \;\, t \in [0, T], \;\;\;\;\;\;\;\;\;\;\;\;\;\;\, \hfill{(\ast)}\\ y(0) &=& y_0 \, \in X, \nonumber \end{eqnarray*} where, $y$ denotes the state space variable, $u$ is the control variable, $A$ is a self adjoint, positive definite linear (not necessarily bounded) operator in a Hilbert space $X$ with dense domain $D(A) \subset X,$ $B(t,s)$ is an unbounded operator, smooth with respect to $t$ and $s$ with $D(A) \subset D(B(t,s)) \subset X$ for $0 \leq s \leq t \leq T$ and $G$ is a bounded linear operator from the control space to $X.$ Assuming that the corresponding evolution equation ($B \equiv 0$ in ($\ast$)) is approximately controllable, it is shown that the set of approximate controls of the distributed control problem ($\ast$) is nonempty. The problem is first viewed as constrained optimal control problem and then it is approximated by unconstrained problem with a suitable penalty function. The optimal pair of the constrained problem is obtained as the limit of optimal pair sequence of the unconstrained problem. The approximation theorems, which guarantee the convergence of the numerical scheme to the optimal pair sequence, are also proved.

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arXiv2015

Optimal Error Estimates for Semidiscrete Galerkin approximations to the Equations of Motion Described by Kelvin-Voigt Viscoelastic Fluid Flow Model

In this paper, the finite element Galerkin method is applied to the equations of motion arising in the Kelvin-Voigt viscoelastic fluid flow model, when the forcing function is in $L^{\infty}(L^2)$. Some a priori estimates for the exact solution, which are valid uniformly in time as $t\mapsto \infty$ and even uniformly in the retardation time $κ$ as $κ\mapsto 0$, are derived. It is shown that the semidiscrete method admits a global attractor. Further, with the help of a priori bounds and Sobolev-Stokes projection, optimal error estimates for the velocity in $L^{\infty}(L^2)$ and $L^{\infty}(H^1_0)$-norms and for the pressure in $L^{\infty}(L^2)$-norm are established. Since the constants involved in error estimates have an exponential growth in time, therefore, in the last part of the article, under certain uniqueness condition, the error bounds are established which are valid uniformly in time. Finally, some numerical experiments are conducted which confirm our theoretical findings.

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arXiv2015

On Kirchhoff's Model of Parabolic Type

In this paper, existence of a strong global solution for all finite time is derived for the Kirchhoff's model of parabolic type. Based on exponential weight function, some new regularity results which reflect the exponential decay property are obtained for the exact solution. For the related dynamics, existence of a global attractor is shown to hold for the problem, when the non- homogeneous forcing function is either independent of time or in $L^{\infty}(L^2)$. With finite element Galerkin method applied in spatial direction keeping time variable continuous, a semidiscrete scheme is analyzed and it is, further, established that the semi-discrete system has a global discrete attractor. Optimal error estimates in $L^{\infty}(H^1_0)$-norm are derived which are valid uniformly in time. Further, based on a Backward Euler method, a completely discrete scheme is developed and error estimates are derived. It is further observed that in case $f = 0$ or $f =O(e^{-γ_0 t})$ with $γ_0 > 0,$ the discrete solutions and also error estimates decay exponentially. Finally, some numerical experiments are discussed which confirm our theoretical findings.

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arXiv2015

Finite volume element method for two-dimensional fractional subdiffusion problems

In this paper, a semi-discrete spatial finite volume (FV) method is proposed and analyzed for approximating solutions of anomalous subdiffusion equations involving a temporal fractional derivative of order $α\in (0,1)$ in a two-dimensional convex polygonal domain. Optimal error estimates in $L^\infty(L^2)$- norm is shown to hold. Superconvergence result is proved and as a consequence, it is established that quasi-optimal order of convergence in $L^{\infty}(L^{\infty})$ holds. We also consider a fully discrete scheme that employs FV method in space, and a piecewise linear discontinuous Galerkin method to discretize in temporal direction. It is, further, shown that convergence rate is of order $O(h^2+k^{1+α}),$ where $h$ denotes the space discretizing parameter and $k$ represents the temporal discretizing parameter. Numerical experiments indicate optimal convergence rates in both time and space, and also illustrate that the imposed regularity assumptions are pessimistic.

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arXiv2014

Convergence of an adaptive mixed finite element method for general second order linear elliptic problems

The convergence of an adaptive mixed finite element method for general second order linear elliptic problems defined on simply connected bounded polygonal domains is analyzed in this paper. The main difficulties in the analysis are posed by the non-symmetric and indefinite form of the problem along with the lack of the orthogonality property in mixed finite element methods. The important tools in the analysis are a posteriori error estimators, quasi-orthogonality property and quasi-discrete reliability established using representation formula for the lowest-order Raviart-Thomas solution in terms of the Crouzeix-Raviart solution of the problem. An adaptive marking in each step for the local refinement is based on the edge residual and volume residual terms of the a posteriori estimator. Numerical experiments confirm the theoretical analysis.

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arXiv2014

A Priori $hp$-estimates for Discontinuous Galerkin Approximations to Linear Hyperbolic Integro-Differential Equations

An $hp$-discontinuous Galerkin (DG) method is applied to a class of second order linear hyperbolic integro-differential equations. Based on the analysis of an expanded mixed type Ritz-Volterra projection, {\it a priori} $hp$-error estimates in $L^{\infty}(L^2)$-norm of the velocity as well as of the displacement, which are optimal in the discretizing parameter $h$ and suboptimal in the degree of polynomial $p$ are derived. For optimal estimates of the displacement in $L^{\infty}(L^2)$-norm with reduced regularity on the exact solution, a variant of Baker's nonstandard energy formulation is developed and analyzed. Results on order of convergence which are similar in spirit to linear elliptic and parabolic problems are established for the semidiscrete case after suitably modifying the numerical fluxes. For the completely discrete scheme, an implicit-in-time procedure is formulated, stability results are derived and {\it a priori} error estimates are discussed. Finally, numerical experiments on two dimensional domains are conducted which confirm the theoretical results.

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arXiv2014

On a three level two-grid finite element method for the 2D-transient Navier-Stokes equations

In this paper, an error analysis of a three steps two level Galekin finite element method for the two dimensional transient Navier-Stokes equations is discussed. First of all, the problem is discretized in spatial direction by employing finite element method on a coarse mesh $\mathcal{T} _H$ with mesh size $H$. Then, in step two, the nonlinear system is linearized around the coarse grid solution, say, $u_H$, which is similar to Newton's type iteration and the resulting linear system is solved on a finer mesh $\mathcal{T}_h$ with mesh size $h$. In step three, a correction is obtained through solving a linear problem on the finer mesh and an updated final solution is derived. Optimal error estimates in $L^{\infty}({\bf L}^2)$-norm, when $h=\mathcal{O} (H^{2-δ})$ and in $L^{\infty}(\bH^1)$-norm, when $h=\mathcal{O}(H^{4-δ})$ for the velocity and in $L^{\infty}(L^2)$-norm, when $h=\mathcal{O}(H^{4-δ})$ for the pressure are established for arbitrarily small $δ>0$. Further, under uniqueness assumption, these estimates are proved to be valid uniformly in time. Then based on backward Euler method, a completely discrete scheme is analyzed and {\it a priori} error estimates are derived. Finally, the paper is concluded with some numerical experiments.

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arXiv2014

Optimal error estimates of mixed FEMs for second order hyperbolic integro-differential equations with minimal smoothness on initial data

In this article, mixed finite element methods are discussed for a class of hyperbolic integro-differential equations (HIDEs). Based on a modification of the nonstandard energy formulation of Baker, both semidiscrete and completely discrete implicit schemes for an extended mixed method are analyzed and optimal L^{\infty}(L^2)-error estimates are derived under minimal smoothness assumptions on the initial data. Further, quasi-optimal estimates are shown to hold in L^{\infty}(L^{\infty})-norm. Finally, the analysis is extended to the standard mixed method for HIDEs and optimal error estimates in L^{\infty}(L^2)-norm are derived again under minimal smoothness on initial data.

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arXiv2014

A priori error estimates for finite volume element approximations to second order linear hyperbolic integro-differential equations

In this paper, both semidiscrete and completely discrete finite volume element methods (FVEMs) are analyzed for approximating solutions of a class of linear hyperbolic integro- differential equations in a two-dimensional convex polygonal domain. The effect of numerical quadrature is also examined. In the semidiscrete case, optimal error estimates in L^{\infty}(L2) and L^{\infty}(H1)- norms are shown to hold with minimal regularity assumptions on the initial data, whereas quasi-optimal estimate in derived in L^{\infty}(L^{\infty})-norm under higher regularity on the data. Based on a second order explicit method in time, a completely discrete scheme is examined and optimal error estimates are established with a mild condition on the space and time discretizing parameters. Finally, some numerical experiments are conducted which confirm the theoretical order of convergence.

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arXiv2014

Error analysis of nonconforming and mixed FEMs for second-order linear non-selfadjoint and indefinite elliptic problems

The state-of-the art proof of a global inf-sup condition on mixed finite element schemes does not allow for an analysis of truly indefinite, second-order linear elliptic PDEs. This paper, therefore, first analyses a nonconforming finite element discretization which converges owing to some a priori $L^2$ error estimates even for reduced regularity on non-convex polygonal domains. An equivalence result of that nonconforming finite element scheme to the mixed finite element method (MFEM) leads to the well-posedness of the discrete solution and to a priori error estimates for the MFEM. The explicit residual-based a posteriori error analysis allows some reliable and efficient error control and motivates some adaptive discretization which improves the empirical convergence rates in three computational benchmarks.

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arXiv2012

Backward Euler method for the Equations of Motion Arising in Oldroyd Fluids of Order One with Nonsmooth Initial Data

In this paper, a backward Euler method is discussed for the equations of motion arising in the 2D Oldroyd model of viscoelastic fluids of order one with the forcing term independent of time or in $L^{\infty}$ in time. It is shown that the estimates of the discrete solution in Dirichlet norm is bounded uniformly in time. Optimal a priori error estimate in L2-norm is derived for the discrete problem with non-smooth initial data. This estimate is shown to be uniform in time, under the assumption of uniqueness condition.

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