Dr Bishnu Lamichhane

© 2018 Walter de Gruyter GmbH, Berlin/Boston 2018 p-Laplace Obstacle Problem. We consider a mixed finite element method for an obstacle problem with the p-Laplace differential operator for p ¿ (1 ,8), where the obstacle condition is imposed by using a Lagrange multiplier. In the discrete setting the Lagrange multiplier basis forms a biorthogonal system with the standard finite element basis so that the variational inequality can be realized in the point-wise form. We provide a general a posteriori error estimate for adaptivity and prove an a priori error estimate. We present numerical results for the adaptive scheme (mesh-size adaptivity with and without polynomial degree adaptation) for the singular case p = 1.5 {p=1.5} and the degenerated case p = 3 {p=3} . We also present numerical results on the mesh independency and on the polynomial degree scaling of the discrete inf-sup constant when using biorthogonal basis functions for the dual variable defined on the same mesh with the same polynomial degree distribution.

© 2018 Elsevier Ltd The frequency-domain and time-domain response of a floating ice shelf to wave forcing are calculated using the finite element method. The boundary conditions at the front of the ice shelf, coupling it to the surrounding fluid, are written as a special non-local linear operator with forcing. This operator allows the computational domain to be restricted to the water cavity beneath the ice shelf. The ice shelf motion is expanded using the in vacuo elastic modes and the method of added mass and damping, commonly used in the hydroelasticity of ships, is employed. The ice shelf is assumed to be of constant thickness while the fluid domain is allowed to vary. The analysis is extended from the frequency domain to the time domain, and the resonant behaviour of the system is studied. It is shown that shelf submergence affects the resonant vibration frequency, whereas the corresponding mode shapes are insensitive to the submergence in constant depth. Further, the modes are shown to have a property of increasing node number with increasing frequency.

© 2018, Springer Science+Business Media, LLC, part of Springer Nature. 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 P1 finite element space and gradient recovery operators. Results of numerical experiments are presented for this novel scheme and for a finite volume scheme.

© 2017 Australian Mathematical Society. An initial-boundary value problem for a time-fractional diffusion equation is discretized in space, using continuous piecewise-linear finite elements on a domain with a re-entrant corner. Known error bounds for the case of a convex domain break down, because the associated Poisson equation is no longer -regular. In particular, the method is no longer second-order accurate if quasi-uniform triangulations are used. We prove that a suitable local mesh refinement about the re-entrant corner restores second-order convergence. In this way, we generalize known results for the classical heat equation.

© 2015, Springer Science+Business Media New York. The thin plate spline method is a widely used data fitting technique as it has the ability to smooth noisy data. Here we consider a mixed finite element discretisation of the thin plate spline. By using mixed finite elements the formulation can be defined in-terms of relatively simple stencils, thus resulting in a system that is sparse and whose size only depends linearly on the number of finite element nodes. The mixed formulation is obtained by introducing the gradient of the corresponding function as an additional unknown. The novel approach taken in this paper is to work 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. Some numerical results are presented to demonstrate the performance of our approach. A preconditioned conjugate gradient method is an efficient solver for the arising linear system of equations.

We consider a mixed finite element method for approximating the solution of nearly incompressible elasticity and Stokes equations. The finite element method is based on quadrilateral and hexahedral triangulation using primal and dual meshes. We use the standard bilinear and trilinear finite element space enriched with element-wise defined bubble functions with respect to the primal mesh for the displacement or velocity, whereas the pressure space is discretized by using a piecewise constant finite element space with respect to the dual mesh. © 2013 Elsevier B.V. All rights reserved.

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. The first approach is based on a so-called standard Lagrange multiplier space for the mortar finite element method, where the Lagrange multiplier basis functions are continuous. The second approach is based on a so-called dual Lagrange multiplier space, where the Lagrange multiplier basis functions are discontinuous. The advantage of using the second approach is that easy static condensation of degrees of freedom corresponding to the Lagrange multiplier is possibly leading to a symmetric positive definite formulation. © 2013 IMACS. Published by Elsevier B.V. All rights reserved.

We consider a mixed finite element method based on simplicial triangulations for a three-field formulation of linear elasticity. The three-field formulation is based on three unknowns: displacement, stress and strain. In order to obtain an efficient discretization scheme, we use a pair of finite element bases forming a biorthogonal system for the strain and stress. The biorthogonality relation allows us to statically condense out the strain and stress from the saddle-point system leading to a symmetric and positive-definite system. The strain and stress can be recovered in a post-processing step simply by inverting a diagonal matrix. Moreover, we show a uniform convergence of the finite element approximation in the incompressible limit. Numerical experiments are presented to support the theoretical results. © 2013 Elsevier B.V.

The uniform convergence of finite element approximations based on a modified Hu-Washizu formulation for the nearly incompressible linear elasticity is analyzed. We show the optimal and robust convergence of the displacement-based discrete formulation in the nearly incompressible case with the choice of approximations based on quadrilateral and hexahedral elements. These choices include bases that are well known, as well as newly constructed bases. Starting from a suitable three-field problem, we extend our a-dependent three-field formulation to geometrically nonlinear elasticity with Saint-Venant Kirchhoff law. Additionally, an a-dependent three-field formulation for a general hyperelastic material model is proposed. A range of numerical examples using different material laws for small and large strain elasticity is presented. © 2007 Elsevier B.V. All rights reserved.