Papers by Bishnu Lamichhane
Australian & New Zealand industrial and applied mathematics journal, Oct 11, 2019
The virtual element method is an extension of the finite element method on polygonal meshes. The ... more The virtual element method is an extension of the finite element method on polygonal meshes. The virtual element basis functions are generally unknown inside an element and suitable projections of the basis functions onto polynomial spaces are used to construct the elemental stiffness and mass matrices. We present a gradient recovery method based on an oblique projection, where the gradient of the L 2polynomial projection of a solution is projected onto a virtual element space. This results in a computationally efficient numerical method. We present numerical results computing the gradients on different polygonal meshes to demonstrate the flexibility of the method.
International Journal of Numerical Analysis and Modeling, 2009
We consider the coupling of compressible and nearly incompressible materials within the framework... more We consider the coupling of compressible and nearly incompressible materials within the framework of mortar methods. Taking into account the locking effect, we use a suitable discretization for the nearly incompressible material and work with a standard conforming discretization elsewhere. The coupling of different discretization schemes in different subdomains are handled by flexible mortar techniques. A priori error analysis is carried out for the coupled problem, and several numerical examples are presented. Using dual Lagrange multipliers, the Lagrange multipliers can easily be eliminated by local static condensation.
arXiv (Cornell University), May 16, 2023
The aim of this research is to examine various statistical approximation properties with respect ... more The aim of this research is to examine various statistical approximation properties with respect to Kantorovich Ó-Baskakov operators using wavelets. We discuss and investigate a weighted statistical approximation employing a Bohman-Korovkin type theorem as well as a statistical rate of convergence applying a weighted modulus of smoothness ω ρα correlated with the space B ρα (R + ) and Lipschitz type maximal functions. Both topics are covered in the article.
arXiv (Cornell University), Sep 4, 2015
We introduce a new minimisation principle for Poisson equation using two variables: the solution ... more We introduce a new minimisation principle for Poisson equation using two variables: the solution and the gradient of the solution. This principle allows us to use any conforming finite element spaces for both variables, where the finite element spaces do not need to satisfy a so-called inf-sup condition. A numerical example demonstrates the superiority of the approach.
Australian & New Zealand industrial and applied mathematics journal, Jan 11, 2016
The thin plate spline method is a widely used data fitting technique which has the ability to smo... more The thin plate spline method is a widely used data fitting technique which has the ability to smooth noisy data. We present some example applications of a new mixed finite element discretisation of the thin plate spline method. The new approach works with a pair of bases for the gradient and the Lagrange multiplier forming a biorthogonal system, thus ensuring that the scheme is numerically efficient and the formulation is stable. We overview of the theoretical foundations of the new approach and give numerical examples in both two and three dimensions.
Australian & New Zealand industrial and applied mathematics journal, May 12, 2013
We propose a novel stabilised mixed finite element method for the discretisation of thin plate sp... more We propose a novel stabilised mixed finite element method for the discretisation of thin plate splines. The mixed formulation is obtained by introducing the gradient of the smoother as an additional unknown. Working with a pair of bases for the gradient of the smoother and the Lagrange multiplier, which forms a biorthogonal system, we eliminate these two variables (gradient of the smoother and Lagrange multiplier) leading to a positive definite formulation. We prove a sub-optimal a priori error estimate for the proposed finite element scheme.
International Journal for Numerical Methods in Fluids, Oct 11, 2013
We present a finite element method for Stokes equations using the Crouzeix-Raviart element for th... more We present a finite element method for Stokes equations using the Crouzeix-Raviart element for the velocity and the continuous linear element for the pressure. We show that the inf-sup condition is satisfied for this pair. Two numerical experiments are presented to support the theoretical results.
Ima Journal of Numerical Analysis, Dec 5, 2017
We consider a saddle-point formulation for a sixth-order partial differential equation and its fi... more We consider a saddle-point formulation for a sixth-order partial differential equation and its finite element approximation, for two sets of boundary conditions. We follow the Ciarlet-Raviart formulation for the biharmonic problem to formulate our saddle-point problem and the finite element method. The new formulation allows us to use the H 1 -conforming Lagrange finite element spaces to approximate the solution. We prove A priori error estimates for our approach. Numerical results are presented for linear and quadratic finite element methods.
Applied Numerical Mathematics, Oct 1, 2013
We present two simple finite element methods for the discretization of Reissner-Mindlin plate equ... more We present two simple finite element methods for the discretization of Reissner-Mindlin plate equations with clamped boundary condition. These finite element methods are based on discrete Lagrange multiplier spaces from mortar finite element techniques. We prove optimal a priori error estimates for both methods.
Journal of Scientific Computing, Aug 3, 2010
We consider a finite element method based on biorthogonal or quasi-biorthogonal systems for the b... more We consider a finite element method based on biorthogonal or quasi-biorthogonal systems for the biharmonic problem. The method is based on the primal mixed finite element method due to Ciarlet and Raviart for the biharmonic equation. Using different finite element spaces for the stream function and vorticity, this approach leads to a formulation only based on the stream function. We prove optimal a priori estimates for both stream function and vorticity, and present numerical results to demonstrate the efficiency of the approach.
Australian & New Zealand industrial and applied mathematics journal, Aug 30, 2020
A wavelength resolved measurement technique used in neutron imaging applications is known as ener... more A wavelength resolved measurement technique used in neutron imaging applications is known as energy-resolved neutron transmission imaging. This technique of reconstructing residual strain maps provides high spatial resolution measurements of strain distribution in polycrystalline materials from sets of Bragg edge measurement images. Strain field reconstructions obtained from both triangular and quadrilateral finite element meshes are compared. The reconstruction is approached via a least square method and relies on the inversion of the longitudinal ray transform, which has uniqueness issues.
Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences, Oct 1, 2021
A solution method is developed for a model of ice shelf vibrations in response to ocean waves, in... more A solution method is developed for a model of ice shelf vibrations in response to ocean waves, in which the ice shelf thickness and seabed beneath the ice shelf vary over distance, and the ice shelf/sub-ice-shelf cavity are connected to the open ocean. The method combines a decomposition of the ice shelf motion into free modes of vibration, a finite element method for the cavity water motion, and a non-local operator to connect to the open ocean. An investigation is conducted into the effects of ice shelf thickening, seabed shoaling and the grounding-line conditions on ice shelf vibrations, induced by regular incident waves in the swell regime. Further, results are given for ice shelf vibrations in response to irregular incident waves, and ocean-to-ice-shelf transfer functions are derived. The findings add to evidence that ice shelves experience appreciable vibrations in response to swell, and that ice shelf thickening and seabed shoaling can have a considerable influence on predictions of how ice shelves respond to ocean waves.
arXiv (Cornell University), Apr 30, 2023
We consider a mixed finite element method for a biharmonic equation with clamped boundary conditi... more We consider a mixed finite element method for a biharmonic equation with clamped boundary conditions based on biorthogonal systems with weakly imposed Dirichlet boundary condition. We show that the weak imposition of the boundary condition arising from a natural minimisation formulation allows to get an optimal a priori error estimate for the finite element scheme improving the existing error estimate for such a formulation without weakly imposed Dirichlet boundary condition. We also briefly outline the algebraic formulation arising from the finite element method.
arXiv (Cornell University), Mar 19, 2018
In this paper, we propose a unified framework, the Hessian discretisation method (HDM), which is ... more In this paper, we propose a unified framework, the Hessian discretisation method (HDM), which is based on four discrete elements (called altogether a Hessian discretisation) and a few intrinsic indicators of accuracy, independent of the considered model. An error estimate is obtained, using only these intrinsic indicators, when the HDM framework is applied to linear fourth order problems. It is shown that HDM encompasses a large number of numerical methods for fourth order elliptic problems: finite element methods (conforming and non-conforming) as well as finite volume methods. We also use the HDM to design a novel method, based on conforming P 1 finite element space and gradient recovery operators. Results of numerical experiments are presented for this novel scheme and for a finite volume scheme.
arXiv (Cornell University), Mar 9, 2021
A solution method is developed for a model of ice shelf vibrations in response to ocean waves, in... more A solution method is developed for a model of ice shelf vibrations in response to ocean waves, in which the ice shelf thickness and seabed beneath the ice shelf vary over distance, and the ice shelf/sub-ice-shelf cavity are connected to the open ocean. The method combines a decomposition of the ice shelf motion into free modes of vibration, a finite element method for the cavity water motion, and a non-local operator to connect to the open ocean. An investigation is conducted into the effects of ice shelf thickening, seabed shoaling and the grounding-line conditions on ice shelf vibrations, induced by regular incident waves in the swell regime. Further, results are given for ice shelf vibrations in response to irregular incident waves, and ocean-to-ice-shelf transfer functions are derived. The findings add to evidence that ice shelves experience appreciable vibrations in response to swell, and that ice shelf thickening and seabed shoaling can have a considerable influence on predictions of how ice shelves respond to ocean waves.
arXiv (Cornell University), Jan 29, 2014
We present a simple finite element method for the discretization of Reissner-Mindlin plate equati... more We present a simple finite element method for the discretization of Reissner-Mindlin plate equations. The finite element method is based on using the nonconforming Crouzeix-Raviart finite element space for the transverse displacement, and the standard linear finite element space for the rotation of the transverse normal vector. We also present two examples for the discrete Lagrange multiplier space for the proposed formulation.
arXiv (Cornell University), Feb 16, 2014
The Gradient Scheme framework provides a unified analysis setting for many different families of ... more The Gradient Scheme framework provides a unified analysis setting for many different families of numerical methods for diffusion equations. We show in this paper that the Gradient Scheme framework can be adapted to elasticity equations, and provides error estimates for linear elasticity and convergence results for non-linear elasticity. We also establish that several classical and modern numerical methods for elasticity are embedded in the Gradient Scheme framework, which allows us to obtain convergence results for these methods in cases where the solution does not satisfy the full H 2 -regularity or for non-linear models.
Australian & New Zealand industrial and applied mathematics journal, Jul 4, 2018
In this paper, we consider an optimal control problem governed by elliptic differential equations... more In this paper, we consider an optimal control problem governed by elliptic differential equations posed in a three-field formulation. Using the gradient as a new unknown we write a weak equation for the gradient using a Lagrange multiplier. We use a biorthogonal system to discretise the gradient, which leads to a very efficient numerical scheme. A numerical example is presented to demonstrate the convergence of the finite element approach.
Australian & New Zealand industrial and applied mathematics journal, Oct 23, 2019
Dye-sensitized solar cells have generated diverse research directions, which include a mathematic... more Dye-sensitized solar cells have generated diverse research directions, which include a mathematical model based on the diffusion of electrons in the conduction band of a nano-porous semiconductor (traditionally TiO 2 ). We solve the nonlinear diffusion equation under its boundary conditions, as stated by Anta et al. [J. Phys. Chem. B 110 (2006) pp 5372-5378]. We employ a standard finite difference method, a fourth order finite difference method scheme and a Runge-Kutta scheme. We calculate errors and evaluate the utility of each scheme as it applies to this boundary value problem.
PLOS Computational Biology, Jul 7, 2021
Locusts are short horned grasshoppers that exhibit two behaviour types depending on their local p... more Locusts are short horned grasshoppers that exhibit two behaviour types depending on their local population density. These are: solitarious, where they will actively avoid other locusts, and gregarious where they will seek them out. It is in this gregarious state that locusts can form massive and destructive flying swarms or plagues. However, these swarms are usually preceded by the aggregation of juvenile wingless locust nymphs. In this paper we attempt to understand how the distribution of food resources affect the group formation process. We do this by introducing a multi-population partial differential equation model that includes non-local locust interactions, local locust and food interactions, and gregarisation. Our results suggest that, food acts to increase the maximum density of locust groups, lowers the percentage of the population that needs to be gregarious for group formation, and decreases both the required density of locusts and time for group formation around an optimal food width. Finally, by looking at foraging efficiency within the numerical experiments we find that there exists a foraging advantage to being gregarious.
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Papers by Bishnu Lamichhane