Amiya K. Pani

• ph.d.(IITK), FNASc,FASc
• Visiting Professor at Birla Institute of Technology and Science, Pilani - Goa Campus

The stability and error analysis of a non-uniform implicit-explicit Alikhanov finite element method (IMEX-Alikhanov-FEM) are investigated for a class of time-fractional linear partial differential/integro-differential equations. These equations involve a non-self-adjoint elliptic operator with variable coefficients in both space and time. A second-...

A non-uniform implicit-explicit L1 mixed finite element method (IMEX-L1-MFEM) is investigated for a class of time-fractional partial integro-differential equations (PIDEs) with space-time-dependent coefficients and non-self-adjoint elliptic part. The proposed fully discrete method combines an IMEX-L1 method on a graded mesh in the temporal variable...

In the first part of this paper, uniqueness of strong solution is established for the Vlasov-unsteady Stokes problem in 3D. The second part deals with a semi discrete scheme, which is based on the coupling of discontinuous Galerkin approximations for the Vlasov and the Stokes equations for the 2D problem. The proposed method is both mass and moment...

This paper deals with a fully discrete numerical scheme for the incompressible Chemotaxis(Keller-Segel)-Navier-Stokes system. Based on a discontinuous Galerkin finite element scheme in the spatial directions, a semi-implicit first-order finite difference method in the temporal direction is applied to derive a completely discrete scheme. With the he...

In this article, a hybridizable discontinuous Galerkin (HDG) method is proposed and analyzed for the Klein-Gordon equation with local Lipschitz-type non-linearity. {\it A priori} error estimates are derived, and it is proved that approximations of the flux and the displacement converge with order $O(h^{k+1}),$ where $h$ is the discretizing paramete...

In this paper, C1-conforming element methods are analyzed for the stream function formulation of a single layer non-stationary quasi-geostrophic equation in the ocean circulation model. In its first part, some new regularity results are derived, which show exponential decay property when the wind shear stress is zero or exponentially decaying. More...

A non-uniform implicit-explicit L1 mixed finite element method (IMEX-L1-MFEM) is investigated for a class of time-fractional partial integro-differential equations (PIDEs) with space-time dependent coefficients and non-self-adjoint elliptic part. The proposed fully discrete method combines an IMEX-L1 method on a graded mesh in the temporal variable...

The stability and error analysis of a non-uniform implicit-explicit Alikhanov finite element method (IMEX-Alikhanov-FEM) are investigated for a class of time-fractional linear partial differential/integro-differential equations. These equations involve a non-self-adjoint elliptic operator with variable coefficients in both space and time. A second-...

Stability and optimal convergence analysis of a non-uniform implicit-explicit L1 finite element method (IMEX-L1-FEM) is studied for a class of time-fractional linear partial differential/integro-differential equations with non-self-adjoint elliptic part having (space-time) variable coefficients. The proposed scheme is based on a combination of an I...

This paper develops and analyses a semi-discrete numerical method for the two-dimensional Vlasov–Stokes system with periodic boundary condition. The method is based on the coupling of the semi-discrete discontinuous Galerkin method for the Vlasov equation with discontinuous Galerkin scheme for the stationary incompressible Stokes equation. The prop...

In this paper, both semi-discrete and fully discrete finite element methods are analyzed for the penalized two-dimensional unsteady Navier–Stokes equations with nonsmooth initial data. First order backward Euler method is applied for the time discretization, whereas conforming finite element method is used for the spatial discretization. Optimal L2...

We formulate and analyze a fully discrete approximate solution of the linear Schrödinger equation on the unit square written as a Schrödinger-type system. The finite element Galerkin method is used for the spatial discretization, and the time stepping is done with an alternating direction implicit extrapolated Crank-Nicolson method. We demonstrate...

By rewriting the Riemann–Liouville fractional derivative as Hadamard finite-part integral and with the help of piecewise quadratic interpolation polynomial approximations, a numerical scheme is developed for approximating the Riemann–Liouville fractional derivative of order $$\alpha \in (1,2).$$ α ∈ ( 1 , 2 ) . The error has the asymptotic expansio...

Approximating the Hadamard finite-part integral by the quadratic interpolation polynomials, we obtain a scheme for approximating the Riemann-Liouville fractional derivative of order $$\alpha \in (1, 2)$$ α ∈ ( 1 , 2 ) and the error is shown to have the asymptotic expansion $$ \big ( d_{3} \tau ^{3- \alpha } + d_{4} \tau ^{4-\alpha } + d_{5} \tau ^{...

The lowest-order nonconforming virtual element extends the Morley triangular element to polygons for the approximation of the weak solution u ∈ V := H20 (Ω) to the biharmonic equation. The abstract framework allows (even a mixture of) two examples of the local discrete spaces Vh(P ) and a smoother allows rough source terms F ∈ V ∗ = H−2(Ω). The a p...

This paper deals with the Vlasov-Stokes' system in three dimensions with periodic boundary conditions in the spatial variable. We prove the existence of a unique strong solution to this two-phase model under the assumption that initial velocity moments of certain order are bounded. We use a fixed point argument to arrive at a global-in-time solutio...

In this paper, semi-discrete numerical scheme for the approximation of the periodic Vlasov-viscous Burgers’ system is developed and analyzed. The scheme is based on the coupling of discontinuous Galerkin approximations for the Vlasov equation and local discontinuous Galerkin approximations for the viscous Burgers’ equation. Both these methods use g...

We formulate and analyze a fully-discrete approximate solution of the linear Schrödinger equation on the unit square written as a Schrödinger-type system. The finite element Galerkin method is used for the spatial discretization, and the time-stepping is done with an alternating direction implicit extrapolated Crank-Nicolson method. We demonstrate...

In this paper, semi-discrete numerical scheme for the approximation of the periodic Vlasov-viscous Burgers' system is developed and analyzed. The scheme is based on the coupling of discontinuous Galerkin approximations for the Vlasov equation and local discontinuous Galerkin approximations for the viscous Burgers' equation. Both these methods use g...

The lowest-order nonconforming virtual element extends the Morley triangular element to polygons for the approximation of the weak solution u ∈ V := H 2 0 (Ω) to the biharmonic equation. The abstract framework allows (even a mixture of) two examples of the local discrete spaces V h (P) and a smoother allows rough source terms F ∈ V * = H −2 (Ω). Th...

Exponential decay estimates of a general linear weakly damped wave equation are studied with decay rate lying in a range. Based on the $C^0$-conforming finite element method to discretize spatial variables keeping temporal variable continuous, a semidiscrete system is analysed, and uniform decay estimates are derived with precisely the same decay r...

Stability and optimal convergence analysis of a non-uniform implicit-explicit L1 finite element method (IMEX-L1-FEM) are studied for a class of time-fractional linear partial differential/integro-differential equations with non-self-adjoint elliptic part having variable (space-time) coefficients. Non-uniform IMEX-L1-FEM is based on a combination of...

In this paper, we consider an optimal control problem governed by linear parabolic differential equations with memory. Under the assumption that the corresponding linear parabolic differential equation without memory term is approximately controllable, it is shown that the set of approximate controls is nonempty. The problem is first viewed as a co...

Residual-based anisotropic a posteriori error estimates are derived for the parabolic integro-differential equation (PIDE) with smooth kernel in two-dimensions. Based on C 0-conforming piecewise linear elements for spatial discretization, the fully discrete method is achieved after discretizing in time by a two-step backward difference (BDF-2) form...

The lowest-order nonconforming virtual element extends the Morley triangular element to polygons for the approximation of the weak solution u ∈ V := H 2 0 (Ω) to the bihar-monic equation. The abstract framework allows (even a mixture of) two examples of the local discrete spaces V h (P) and a smoother allows rough source terms F ∈ V * = H −2 (Ω). T...

In this article, a second order quasi-linear parabolic initial-boundary value problem is approximated by using primal hybrid finite element method and Lagrange multipliers. Semidiscrete and backward Euler based fully discrete schemes are discussed and optimal order error estimates are established by applying modified elliptic projection. Optimal or...

The nonconforming virtual element method (NCVEM) for the approximation of the weak solution to a general linear second-order non-selfadjoint indefinite elliptic PDE in a polygonal domain $\Omega$ is analyzed under reduced elliptic regularity. The main tool in the a priori error analysis is the connection between the nonconforming virtual element sp...

The lowest-order nonconforming virtual element extends the Morley triangular element to polygons for the approximation of the weak solution u ∈ V := H 2 0 (Ω) to the bihar-monic equation. The abstract framework allows (even a mixture of) two examples of the local discrete spaces V h (P) and a smoother allows rough source terms F ∈ V * = H −2 (Ω). T...

The lowest-order nonconforming virtual element extends the Morley triangular element to polygons for the approximation of the weak solution $u\in V:=H^2_0(\Omega)$ to the biharmonic equation. The abstract framework allows (even a mixture of) two examples of the local discrete spaces $V_h(P)$ and a smoother allows rough source terms $F\in V^*=H^{-2}...

For a well-posed non-selfadjoint indefinite second-order linear elliptic PDE with general coefficients A, b, γ in L 8 and symmetric and uniformly positive definite coefficient matrix A, this paper proves that mixed finite element problems are uniquely solvable and the discrete solutions are uniformly bounded, whenever the underlying shape-regular t...

A direct method of identification of time dependent parameters in a linear parabolic boundary value problem with over-specified total internal energy involves the flux at the boundary, and an H 1 mixed formulation seems to be more suitable than the standard methods for such class of nonlocal problems. Therefore, this paper develops and analyses an...

For a well-posed non-selfadjoint indefinite second-order linear elliptic PDE with general coefficients $\mathbf A, \mathbf b,\gamma$ in $L^\infty$ and symmetric and uniformly positive definite coefficient matrix $\mathbf A$, this paper proves that mixed finite element problems are uniquely solvable and the discrete solutions are uniformly bounded,...

In this paper, both semidiscrete and fully discrete finite element methods are analyzed for the penalized two-dimensional unsteady Navier-Stokes equations with nonsmooth initial data. First order backward Euler method is applied for the time discretization, whereas conforming finite element method is used for the spatial discretization. Optimal $L^...

In this paper, both semidiscrete and fully discrete finite element methods are analyzed for the penalized two-dimensional unsteady Navier-Stokes equations with nonsmooth initial data. First order backward Euler method is applied for the time discretization, whereas conforming finite element method is used for the spatial discretization. Optimal L 2...

In this paper, a penalty formulation is proposed and analyzed in both continuous and finite element setups, for the two-dimensional Oldroyd model of order one, when the initial velocity is in H 0 1 . New regularity results which are valid uniformly in time as t → ∞ and in the penalty parameter 𝜀 as ε → 0 are derived for the solution of the penalize...

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 two-dimensional 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 soluti...

The conforming finite element Galerkin method is applied to discretise in the spatial direction for a class of strongly nonlinear parabolic problems. Using elliptic projection of the associated linearised stationary problem with Gronwall type result, optimal error estimates are derived, when piecewise polynomials of degree r≥1 are used, which impro...

Nonconforming Morley finite element method is applied to a fourth order nonlinear reaction-diffusion problems. After deriving some regularity results to be used subsequently in our error analysis, Morley FEM is employed to discretize in the spatial direction to obtain a semidiscrete problem. A priori bounds for the discrete solution are derived and...

A variational formulation is analysed for the Oseen equations written in terms of vorticity 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 pol...

Nonconforming Morley finite element method is applied to a fourth order nonlinear reaction-diffusion problems. After deriving some regularity results to be used subsequently in our error analysis, Morley FEM is employed to discretize in the spatial direction to obtain a semidiscrete problem. A priori bounds for the discrete solution are derived and...

This paper focusses on the von Karman equations for the moderately large deformation of a very thin plate with the convex obstacle constraint leading to a coupled system of semilinear fourth-order obstacle problem and motivates its nonconforming Morley finite element approximation. The first part establishes the well-posedness of the von Karman obs...

The conforming finite element Galerkin method is applied to discretise in the spatial direction for a class of strongly nonlinear parabolic problems. Using elliptic projection of the associated linearised stationary problem with Gronwall type result, optimal error estimates are derived, when piecewise polynomials of degree r ≥ 1 are used, which imp...

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 ∞ in time. It is shown that the estimates of the discrete solution in Dirichlet norm is...

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, a priori optima...

A Galerkin finite element method is applied to approximate the solution of a semilinear stochastic space and time fractional subdiffusion problem with the Caputo fractional derivative of the order $ \alpha \in (0, 1)$, driven by fractionally integrated additive noise. After discussing the existence, uniqueness and regularity results, we approximate...

In this paper, we propose and analyze a nonconforming Morley finite element method for the stationary quasi-geostrophic equation in the ocean circulation. Stability and the inf–sup condition for the discrete solution are proved, and the local existence of a unique solution to the discrete nonlinear system is established based on the assumption of t...

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 N\'ed\'elec finite elements and piecewise cont...

The nonconforming virtual element method (NCVEM) for the approximation of the weak solution to a general linear second-order non-selfadjoint indefinite elliptic PDE in a polygonal domain Ω is analyzed under reduced elliptic regularity. The main tool in the a priori error analysis is the connection between the nonconforming virtual element space and...

In this paper, we propose and analyze a nonconforming Morley finite element method for the stationary quasi-geostrophic equation in the ocean circulation. Stability and the inf-sup condition for the discrete solution are proved, and the local existence of a unique solution to the discrete nonlinear system is established based on the assumption of t...

This paper focusses on the von K\'{a}rm\'{a}n equations for the moderately large deformation of a very thin plate with the convex obstacle constraint leading to a coupled system of semilinear fourth-order obstacle problem and motivates its nonconforming Morley finite element approximation. The first part establishes the well-posedness of the von K\...

In this article, global stabilization results for the two dimensional viscous Burgers’ equation, that is, convergence of unsteady solution to its constant steady state solution with any initial data, are established using a nonlinear Neumann boundary feedback control law. Then, applying \(C^0\)-conforming finite element method in spatial direction,...

We analyze a second order in space, first order in time accurate finite difference method for a spatially periodic convection-diffusion problem. This method is a time stepping method based on the first order Lie splitting of the spatially semidiscrete solution. In each time step, on an interval of length k, of this solution, the method uses the bac...

In this paper, a mixed finite element method is applied in spatial directions while keeping time variable continuous to a class of time-fractional diffusion problems with time-dependent coefficients on a bounded convex polygonal domain. Based on an energy argument combined with a repeated application of an integral operator, optimal error estimates...

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 $BDM_{k+1}$ elemen...

In this paper, a mixed finite element method is applied in spatial directions while keeping time variable continuous to a class of time-fractional diffusion problems with time-dependent coefficients on a bounded convex polygonal domain. Based on an energy argument combined with a repeated application of an integral operator, optimal error estimates...

Two higher order time stepping methods for solving subdiffusion problems are studied in this paper. The Caputo time fractional derivatives are approximated by using the weighted and shifted Grünwald–Letnikov formulae introduced in Tian et al. (Math Comput 84:2703–2727, 2015). After correcting a few starting steps, the proposed time stepping methods...

For a nonlinear nonlocal parabolic problem containing the elastic energy coefficients, an expanded mixed finite element method using lowest order RT spaces is discussed in this paper. Firstly, some new regularity results are derived avoiding compatibility conditions on the data, which reflect behavior of exact solution as t → 0. Then, a semidiscret...

Quasi‐optimal error estimates are derived for the continuous‐time orthogonal spline collocation (OSC) method and also two discrete‐time OSC methods for approximating the solution of 1D parabolic singularly perturbed reaction–diffusion problems. OSC with C1 splines of degree r ≥ 3 on a Shishkin mesh is employed for the spatial discretization while t...

In this article, global stabilization results for the Benjamin–Bona–Mahony–Burgers’ (BBM–B) type equations are obtained using nonlinear Neumann boundary feedback control laws. Based on the \(C^0\)-conforming finite element method, global stabilization results for the semidiscrete solution are also discussed. Optimal error estimates in \(L^\infty (L...

Global stabilization of viscous Burgers' equation around constant steady state solution has been discussed in the literature. The main objective of this paper is to show global stabilization results for the 2D forced viscous Burgers' equation around a nonconstant steady state solution using nonlinear Neumann boundary feedback control law, under som...

An orthogonal spline collocation method (OSCM) with C1 splines of degree r ≥ 3 is analyzed for the numerical solution of singularly perturbed reaction diffusion problems in one dimension. The method is applied on a Shishkin mesh and quasi-optimal error estimates in weighted H m norms for m = 1,2 and in a discrete L ² -norm are derived. These estima...

In this article, global stabilization results for the two dimensional viscous Burgers' equation that is, convergence of unsteady solution to its constant steady state solution with any initial data, are established using a nonlinear Neumann boundary feedback control law. Then, using $C^0$-conforming finite element method, global stabilization resul...

In this article, global stabilization results for the Benjamin-Bona-Mahony-Burgers' (BBM-B) type equations are obtained using nonlinear Neumann boundary feedback control laws. Based on the $C^0$-conforming finite element method, global stabilization results for the semidiscrete solution are also discussed. Optimal error estimates in $L^\infty(L^2)$...

In this article, we discuss error estimates for nonlinear parabolic problems using discontinuous Galerkin methods which include HDG method in the spatial direction while keeping time variable continuous. When piecewise polynomials of degree k ≥ 1 are used to approximate both the potential as well as the flux, it is shown that the error estimate for...

In this paper, an expanded mixed finite element method with lowest order Raviart Thomas elements is developed and analyzed for a class of nonlinear and nonlocal parabolic problems. After obtaining some regularity results for the exact solution, a priori error estimates for the semidiscrete problem are established. Based on a linearized backward Eul...

Three first-order finite volume element methods; namely, the conforming, noncon-forming and discontinuous Galerkin schemes for Stokes equations are analysed and compared employing the medius analysis. The latter is based on the combination of arguments from a priori and a posteriori error analyses under no extra regularity assumptions on the weak s...

In this article, a qualocation method is formulated and analyzed for parabolic partial integro-differential equations in one space variable. Using a new Ritz-Volterra type projection, optimal rates of convergence are derived. Based on the second-order backward differentiation formula, a fully discrete scheme is formulated and a convergence analysis...

Three first-order finite volume element methods; namely, the conforming, noncon-forming and discontinuous Galerkin schemes for Stokes equations are analysed and compared employing the medius analysis. The latter is based on the combination of arguments from a priori and a posteriori error analyses under no extra regularity assumptions on the weak s...

Abstract. In this paper, a finite volume element (FVE) method is considered for spatial approxi-mations of time fractional diffusion equations involving a Riemann-Liouville fractional derivative oforderα∈(0,1) in time. Improving upon earlier results (Karaaet al., IMA J. Numer. Anal. 2016),optimal error estimates inL2(Ω)- andH1(Ω)-norms for the semi...

In this article, stabilization result for the viscoelastic fluid flow problem is 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 convergenc...

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 r...

In this article, we discuss global stabilization results for the Burgers' equation using nonlinear Neumann boundary feedback control law. As a result of the nonlinear feedback control, a typical nonlinear problem is derived. Then, based on C 0-conforming finite element method, global stabilization results for the semidiscrete solution are analyzed....

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, a priori optima...

For fully nonlinear elliptic boundary value problems in divergence form, improved error estimates are derived in the frame work of a class of expanded discontinuous Galerkin methods. It is shown that the error estimate for the discrete flux in $L^2$ -norm is of order $k+1$, when piecewise polynomials of degree $k\geq 1$ are used to approximate both...

In this article, global stabilization results for the Benjamin-Bona-Mahony-Burgers (BBM-B) type equations are established using nonlinear Neumann boundary feedback control laws. Then, based on C 0-conforming finite element method global stabilization results for the semidiscrete solution are discussed. Optimal error estimates in L ∞ (L 2), L ∞ (H 1...

In this paper, stabilization result for the Benjamin-Bona-Mahony-Burgers (BBMB) equation, that is, convergence of unsteady solution to steady state solution is established under the assumption that a linearized steady state eigenvalue problem has a minimal positive eigenvalue. Further, the exponential decay property is shown in L ∞ (H j), j = 0, 1,...

Lecture slides on DGM for elliptic problems

In this article, the convergence of the solution of the Kelvin-Voigt viscoelastic fluid flow model to its steady state solution with exponential rate is established under the uniqueness assumption. Then, a semidiscrete Galerkin method for spatial direction keeping time variable continuous is considered and asymptotic behavior of the semidiscrete so...

In this paper, an expanded mixed finite element method with lowest order Raviart Thomas elements is developed and analyzed for a class of nonlinear and nonlocal parabolic problems. After obtaining some regularity results for the exact solution, a priori error estimates for the semidiscrete problem are established. Based on a linearized backward Eul...

Talk in the International conference :ECM2017

In this paper, we discuss a new stabilized Lagrange multiplier method for finite element solution of multi-domain elliptic and parabolic initial-boundary value problems with non-matching grid across the subdomain interfaces. The proposed method is consistent with the original problem and its stability is established without using the inf-sup (well...

For fully nonlinear elliptic boundary value problems in divergence form, improved error estimates are derived in the frame work of a class of expanded discontinuous Galerkin methods. It is shown that the error estimate for the discrete flux in L 2-norm is of order k + 1, when piecewise polynomials of degree k ≥ 1 are used to approximate both potent...