Linear Finite Element Scheme Research Articles

Two linear energy stable lumped mass finite element schemes for the viscous Cahn–Hilliard equation on curved surfaces in 3D

The evolution of a dynamic system on complex curved 3D surfaces is essential for the understanding of natural phenomena, the development of new materials, and engineering design optimization. In this work, we study the viscous Cahn–Hilliard equation on curved surfaces and develop two linear energy stable finite element schemes based on the lumped mass method. Two stabilizing terms are added to ensure both the unique solvability and unconditional energy stability. We prove rigorously that two schemes are unconditionally energy stable . Numerical experiments are presented to verify theoretical results and to show the robustness and accuracy of the proposed method.

Two second-order and linear numerical schemes for the multi-dimensional nonlinear time-fractional Schrödinger equation

This paper presents two second-order and linear finite element schemes for the multi-dimensional nonlinear time-fractional Schrodinger equation. In the first numerical scheme, we adopt the L2-1σ formula to approximate the Caputo derivative. However, this scheme requires storing the numerical solution at all previous time steps. In order to overcome this drawback, we develop the $\mathcal {F}L2$ -1σ formula to construct the second numerical scheme, which reduces the computational storage and cost. We prove that both the L2-1σ and $\mathcal {F}L2$ -1σ formulas satisfy the three assumptions of the generalized discrete fractional Gronwall inequality. Furthermore, combining with the temporal-spatial error splitting argument, we rigorously prove the unconditional stability and optimal error estimates of these two numerical schemes, which do not require any time-step restrictions dependent on the spatial mesh size. Numerical examples in two and three dimensions are given to illustrate our theoretical results and show that the second scheme based on $\mathcal {F}L2$ -1σ formula can reduce CPU time significantly compared with the first scheme based on L2-1σ formula.

Weak martingale solutions to the stochastic Landau–Lifshitz–Gilbert equation with multi-dimensional noise via a convergent finite-element scheme

We propose an unconditionally convergent linear finite element scheme for the stochastic Landau–Lifshitz–Gilbert (LLG) equation with multi-dimensional noise. By using the Doss–Sussmann technique, we first transform the stochastic LLG equation into a partial differential equation that depends on the solution of the auxiliary equation for the diffusion part. The resulting equation has solutions absolutely continuous with respect to time. We then propose a convergent θ-linear scheme for the numerical solution of the reformulated equation. As a consequence, we are able to show the existence of weak martingale solutions to the stochastic LLG equation.

A combinational efficient algorithm for elliptic eigenvalue problems

In this paper, we present a combinational efficient algorithm for solving second-order elliptic ordinary differential equation eigenvalue problems on four kinds of meshes. Firstly, a combinational scheme is proposed by leveraging the linear finite element scheme and the second-order finite difference scheme based on mass lumping method. An efficient algorithm is constructed by using the combinational scheme with a constant combined parameter. Then, we improve the efficient algorithm by computing the quasi-optimal combined parameters for different eigenvalues. Finally, numerical experiments tested on both uniform and nonuniform meshes are shown to illustrate the efficiency of our algorithms.

A Priori Error Estimates of C rank–Nicolson Finite Volume Element Method for a Hyperbolic Optimal Control Problem

In this article, a Crank–Nicolson linear finite volume element scheme is developed to solve a hyperbolic optimal control problem. We use the variational discretization technique for the approximation of the control variable. The optimal convergent order O(h2 + k2) is proved for the numerical solution of the control, state and adjoint-state in a discrete L2-norm. To derive this result, we also get the error estimate (convergent order O(h2 + k2)) of Crank–Nicolson finite volume element approximation for the second-order hyperbolic initial boundary value problem. Numerical experiments are presented to verify the theoretical results.© 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1331–1356, 2016

Poroelasticity problem: numerical difficulties and efficient multigrid solution

This work contains some of the more relevant results obtained by the author regarding the numerical solution of the Biot’s consolidation problem. The emphasis here is on the stable discretization and the highly efficient solution of the resulting algebraic system of equations, which is of saddle point type. On the one hand, a stabilized linear finite element scheme providing oscillation-free solutions for this model is proposed and theoretically analyzed. On the other hand, a monolithic multigrid method is considered for the solution of the resulting system of equations after discretization by using the stabilized scheme. Since this system is of saddle point type, special smoothers of “Vanka”-type have to be considered. This multigrid method is designed with the help of an special local Fourier analysis that takes into account the specific characteristics of the considered block-relaxations. Results from this analysis are presented and compared with those experimentally computed.

A high order composite scheme for the second order elliptic problem with nonlocal boundary and its fast algorithm

The elliptic problem with nonlocal boundary condition is widely applied in the field of science and engineering. Firstly, we construct a linear finite element scheme for the nonlocal boundary problem, and derive the optimal L2 error estimate. Then, based on the quadratic finite element and the extrapolation linear finite element methods, we present a composite scheme, and prove that it is convergent order three. Furthermore, we design an upper triangular preconditioning algorithm for the linear finite element discrete system. Finally, numerical results not only validate that the new algorithm is efficient, but also show that the new scheme is convergent order three, furthermore order four on uniform grids.

A three level finite element approximation of a pattern formation model in developmental biology

This paper concerns a second-order, three level piecewise linear finite element scheme 2-SBDF (Ruuth, in J Math Biol 34:148–176, 1995 ) for approximating the stationary (Turing) patterns of a well-...

A linear mixed finite element scheme for a nematic Ericksen–Leslie liquid crystal model

In this work we study a fully discrete mixed scheme, based on continuous finite elements in space and a linear semi-implicit first-order integration in time, approximating an Ericksen–Leslie nematic liquid crystal model by means of a Ginzburg–Landau penalized problem. Conditional stability of this scheme is proved via a discrete version of the energy law satisfied by the continuous problem, and conditional convergence towards generalized Young measure-valued solutions to the Ericksen–Leslie problem is showed when the discrete parameters (in time and space) and the penalty parameter go to zero at the same time. Finally, we will show some numerical experiences for a phenomenon of annihilation of singularities.

Some error estimates on the finite element approximation for two-dimensional elliptic problem with nonlocal boundary

We construct a linear finite element scheme for the two-dimensional elliptic problem with nonlocal boundary conditions. In order to obtain the optimal error estimates in the L2 norm, we innovatively decompose the original elliptic problem into two subproblems by the principle of superposition for the differential equation: nonhomogeneous one with Dirichlet boundary and homogeneous one with nonlocal boundary. Then, we prove that our approximate solution has saturated convergent order by properly introducing a projection operator and the maximum principle. In addition, we design an attractive preconditioning algorithm for our discrete system, which improves the efficiency of our computation. Finally, numerical experiments verify our theoretical results.

Finite Element Multigrid Method for the Boundary Value Problem of Fractional Advection Dispersion Equation

The stationary fractional advection dispersion equation is discretized by linear finite element scheme, and a full V-cycle multigrid method (FV-MGM) is proposed to solve the resulting system. Some useful properties of the approximation and smoothing operators are proved. Using these properties we derive the convergence results in both <svg style="vertical-align:-0.0pt;width:16.1625px;" id="M1" height="15.8375" version="1.1" viewBox="0 0 16.1625 15.8375" width="16.1625" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,15.775)"><path id="x1D43F" d="M559 163q-23 -66 -68 -163h-474l6 26q62 4 79.5 19.5t28.5 75.5l78 409q7 35 8.5 49t-8 25t-24 13t-51.5 5l5 28h266l-6 -28q-65 -5 -79.5 -18t-25.5 -74l-76 -406q-10 -57 14 -75q12 -13 96 -13q93 0 126 29q41 40 76 109z" /></g> <g transform="matrix(.012,-0,0,-.012,9.763,7.613)"><path id="x32" d="M412 140l28 -9q0 -2 -35 -131h-373v23q112 112 161 170q59 70 92 127t33 115q0 63 -31 98t-86 35q-75 0 -137 -93l-22 20l57 81q55 59 135 59q69 0 118.5 -46.5t49.5 -122.5q0 -62 -29.5 -114t-102.5 -130l-141 -149h186q42 0 58.5 10.5t38.5 56.5z" /></g> </svg> norm and energy norm for FV-MGM. Numerical examples are given to demonstrate the convergence rate and efficiency of the method.

Node-pair finite volume/finite element schemes for the Euler equation in cylindrical and spherical coordinates

A numerical scheme is presented for the solution of the compressible Euler equations in both cylindrical and spherical coordinates. The unstructured grid solver is based on a mixed finite volume/finite element approach. Equivalence conditions linking the node-centered finite volume and the linear Lagrangian finite element scheme over unstructured grids are reported and used to devise a common framework for solving the discrete Euler equations in both the cylindrical and the spherical reference systems. Numerical simulations are presented for the explosion and implosion problems with spherical symmetry, which are solved in both the axial–radial cylindrical coordinates and the radial–azimuthal spherical coordinates. Numerical results are found to be in good agreement with one-dimensional simulations over a fine mesh.

The Galerkin continuous finite element method for delay-differential equation with a variable term

In this paper, we introduce and analyze the continuous finite element method for a class of delay-differential equation with a variable term. At first for the case of the constant delay and based on an orthogonal expansion in an element we derive optimal superconvergence u − U = O( h p+2 ), p ⩾ 2 at the ( p + 1)-order Lobatto points and u − U = O( h 2 p ) at the integer nodal points. Next we prove the stability of linear and quadratic finite element schemes for the case of constant delay. For the case of variable delay, by use of the approximative to the continuous delay in every subinterval we decompose the original problem into a series of subproblems with constant delay. Finally we give a recursive schemes which can be compute by every subproblem.

Numerical methods for Stefan problems with prescribed convection and nonlinear flux

This paper deals with numerical methods for a class of nonlinear degenerate parabolic equations with nonlinear flux boundary conditions, which in particular includes the well-known continuous casting Stefan problem. A practical piecewise linear finite element scheme with numerical integration which is based on an implicit discretization of the diffusion and an explicit approximation of the enthalpy-dependent convection is proposed. The convergence of the scheme is proved without requiring the quasi-uniformity and the acuteness of the finite element meshes. The key ingredient of the analysis is an estimate on the fractional derivative in time of the discrete temperature which provides the necessary strong convergence property. The results of some numerical tests are provided in order to show the accuracy of the scheme.

Convergence of votex methods in a bounded domain using linear finite elements

In this work, the vortex method for the initial boundary value problem of Euler's equation is considered. We analyze linear finite element methods to solve the Poisson equation, which require less operations to compute the velocity field from the vorticity. We prove that the rate of convergence of the linear finite element scheme is independent of the vortex blob parameters.

Spurious numerical refraction

AbstractThe purpose of this paper is to investigate the effect of a non‐uniform mesh in two dimensions (2D). A change in mesh size will, in general, result in spurious refraction (and reflection) which is entirely numerical (rather than physical) in origin. To facilitate the analysis, the mesh geometry has been highly simplified in that only a single change in mesh size is considered. The analysis is based on a finite element wave model.The domain consists of two conterminous regions discernible only by their different nodal spacings in the x‐direction. The interface between the two regions is internal to the mesh and is a straight line. The model is based upon the Crank‐Nicolson linear finite element scheme applied to the second order wave equation. The results of the analysis are confirmed by numerical experiments. It is shown that under particular numerical conditions total internal reflection may occur and when this is the case, the transmitted wave is evanescent. An analysis of the energy flux associated with the incident, reflected and trasmitted waves shows that energy is conserved across the interface between the two regions.

Internal wave reflections and transmissions arising from a non‐uniform mesh. Part III: The occurrence of evanescent waves in the Crank‐Nicolson linear finite element scheme

AbstractThe numerical scheme upon which this paper is based is the 1D Crank–Nicolson linear finite element scheme. In Part I of this series it was shown that for a certain range of incident wavelengths impinging on the interface of an expansion in nodal spacing, an evanescent (or spatially damped) wave results in the downstream region. Here in Part III an analysis is carried out to predict the wavelength and the spatial rate of damping for this wave. The results of the analysis are verified quantitatively with seven ‘hot‐start’ numerical experiments and qualitatively with seven ‘cold‐start’ experiments. Weare has shown that evanescent waves occur whenever the frequency of a disturbance at a boundary exceeds the maximum frequency given by the dispersion relation. In these circumstances the ‘extended dispersion’ relation can be used to determine the rate of spatial decay.In the context of a domain consisting of two regions with different nodal spacings, the use of the group velocity concept shows that evanescent waves have no energy flux associated with them when energy is conserved.

Internal wave reflections and transmissions arising from a non‐uniform mesh. Part II: A generalized analysis for the Crank–Nicolson linear finite element scheme

AbstractInternal wave reflections and transmissions are examined for the Crank–Nicolson linear finite element scheme applied to the linear shallow water equations in a ID domain containing an abrupt change in nodal spacing. In Part I of this series the reflection/transmission analysis was verified by some ‘hot‐start’ numerical experiments. Here in Part II, however, that analysis is found wanting when it comes to providing a description of the pseudo‐steady state wave configuration which develops with some ‘cold‐start’ experiments. It is shown that the analysis of Part I can be extended to take in both the ‘hot‐’ and ‘cold‐start’ experimental results such that four essentially different wave configurations can be identified. The four configurations are discernible on the basis of group velocity. In order to be sustained, two of the configurations require one energy source whereas the other two require two energy sòurces. Numerical experiments confirmed the analysis.

Internal wave reflections and transmissions arising from a non‐uniform mesh. Part I: An analysis for the Crank–Nicolson linear finite element scheme

AbstractThis is the first of a series of three related papers dealing with some of the consequences of non‐uniform meshes in a numerical model. In this paper the accuracy of the Crank–Nicolson linear finite element scheme, which is applied to the linear shallow water equations, is examined in the context of a single abrupt change in nodal spacing. The (in)accuracy is quantified in terms of reflection and transmission coefficients. An incident wave impinging on the interface between two regions with different nodal spacings is shown to give rise to no reflected waves and two transmitted waves. The analysis is verified using three different wavelengths (2Δx, 4Δx 8Δx) in three ‘hot‐start’ numerical experiments with a mesh expansion factor of 2 and three experiments with a mesh contraction factor of 1/2. An energy flux analysis based on the concept of group velocity shows that energy is conserved across the interface.

Error estimates for two-phase stefan problems in several space variables, II: Non-linear flux conditions

In this paper we derive L2-error estimates for multidimensional two-phase Stefan problems involving further non-linearities such as non-linear flux conditions. We approximate the enthalpy formulation by a regularization procedure combined with a C0-piecewise linear finite element scheme in space, and the implicit Euler scheme in time.