L2 Error Estimates Research Articles

A new two-grid mixed finite element analysis of semi-linear reaction–diffusion equation

A new mixed finite element method is established for solving the semi-linear reaction–diffusion equation based on two-grid discretization. Here, a symmetric positive definite mixed finite element method is constructed for semi-linear problem, and the corresponding error estimates in L2 and Lp-norms are considered. Then, the two-grid techniques with and without one Newton iteration correction are applied for the proposed mixed element procedure to improve the efficiency of computation. The corresponding convergence of the proposed algorithms is analyzed, and the asymptotically optimal approximation is achieved with the relation H=O(h12). Finally, numerical examples are provided to confirm theoretical results.

Galerkin finite element methods for the numerical solution of two classical-Boussinesq type systems over variable bottom topography

We consider two ‘Classical’ Boussinesq type systems modelling two-way propagation of long surface waves in a finite channel with variable bottom topography. Both systems are derived from the 1-d Serre–Green–Naghdi (SGN) system; one of them is valid for stronger bottom variations, and coincides with Peregrine’s system, and the other is valid for smaller bottom variations. We discretize in the spatial variable simple initial–boundary-value problems (ibvp’s) for both systems using standard Galerkin-finite element methods and prove L2 error estimates for the ensuing semidiscrete approximations. We couple the schemes with the 4th order-accurate, explicit, classical Runge–Kutta time-stepping procedure and use the resulting fully discrete methods in numerical simulations of dispersive wave propagation over variable bottoms with several kinds of boundary conditions, including absorbing ones. We describe in detail the changes that solitary waves undergo when evolving under each system over a variety of variable-bottom environments. We assess the efficacy of both systems in approximating these flows by comparing the results of their simulations with each other, with simulations of the SGN-system, and with available experimental data from the literature.

Unconditionally optimal error estimates of BDF2 Galerkin method for semilinear parabolic equation

AbstractIn this paper, a 2‐step backward differentiation formula (BDF2) Galerkin method is investigated for semilinear parabolic equation. More precisely, the second‐order time‐stepping scheme is used for time discretization and the piecewise linear continuous Galerkin method is employed for spatial discretization, respectively. Optimal error estimates in L2 and H1‐norms are obtained without any restriction on the time‐step size, while previous works always require certain conditions on time step‐size. The key to our analysis is to derive a uniform boundness of the numerical solution in energy norm so as to avoid the inverse inequality used in the usual convergence analysis of the finite element methods. Numerical experiments are carried out to confirm the theoretical analysis.

An accuracy analysis of Galerkin meshfree methods accounting for numerical integration

Presently, it has been widely recognized that numerical integration plays a crucial role on the solution accuracy of Galerkin meshfree methods due to the rational nature of meshfree shape functions and the misalignment of shape function supports with integration cells. Nonetheless, the corresponding underlying cause is still not quite clear. To address this issue, an accuracy analysis of Galerkin meshfree methods is presented in this study with a special focus on the contribution of numerical integration to the error estimates. One important fact is that the integration difficulty of meshfree methods leads to a loss of the Galerkin orthogonality condition which usually serves as a basis to develop error bounds. In order to enable the accuracy analysis, an error measure is proposed to evaluate the loss of Galerkin orthogonality condition, which is actually related to the order of integration consistency for Galerkin meshfree formulation. With the aid of this orthogonality error measure, both H1 and L2 error estimates are established for Galerkin meshfree methods. It turns out that these errors essentially consist of two parts, one comes from the standard interpolation error, and the other one attributes to the numerical integration error. In accordance with the proposed error estimates, it is evident that for the conventional Gauss integration schemes violating the integration consistency, the solution errors eventually are controlled by the integration error and optimal convergence cannot be achieved even with high order quadrature rules. On the other hand, for the consistent integration approaches, e.g., the stabilized conforming nodal integration and the more recent reproducing kernel gradient smoothing integration, the integration and interpolation errors have the same accuracy order and thus an optimal convergence of Galerkin meshfree methods is ensured. The proposed theoretical error estimates for Galerkin meshfree methods are well demonstrated by numerical results.

Error estimates in weak Galerkin finite element methods for parabolic equations under low regularity assumptions

In this paper, we consider the weak Galerkin finite element approximations of second order linear parabolic problems in two dimensional convex polygonal domains under the low regularities of the solutions. Optimal order error estimates in L2(L2) and L2(H1) norms are shown to hold for both the spatially discrete continuous time and the discrete time weak Galerkin finite element schemes, which allow using the discontinuous piecewise polynomials on finite element partitions with the arbitrary shape of polygons with certain shape regularity. The fully discrete scheme is based on first order in time Euler method. We have derived O(hr+1) in L2(L2) norm and O(hr) in L2(H1) norm when the exact solution u∈L2(0,T;Hr+1(Ω))∩H1(0,T;Hr−1(Ω)), for some r≥1. Numerical experiments are reported for several test cases to justify our theoretical convergence results.

A divergence-free reconstruction of the nonconforming virtual element method for the Stokes problem

In this paper, we propose and investigate a divergence-free reconstruction of the nonconforming virtual element for the Stokes problem. By constructing the computable Raviart–Thomas-like interpolation operator, we guarantee the independence between the velocity error estimation |u−uh|1,h and the continuous pressure p, as it happens for the divergence-free flow solver. Moreover, this modified scheme can also inherit the advantages of the classical nonconforming virtual element method, such as, very general meshes including non-convex and degenerate elements, a unified scheme for an arbitrary-order approximation accuracy k, etc. Then, we provide the optimal L2-error estimates for the velocity gradient and the pressure by taking advantage of the Raviart–Thomas-like interpolation operator and avoiding the use of a trace inequality. Finally, three numerical experiments are presented to conform the theoretical analysis.

Unconditional optimal error estimates of linearized second-order BDF Galerkin FEMs for the Landau-Lifshitz equation

In this paper, we establish the unconditionally optimal error estimates of linearized second-order backward difference formula (BDF2) Galerkin finite element methods (FEMs) for the Landau-Lifshitz equation describing the magnetic behavior in ferromagnetic materials. By using the temporal-spatial error splitting techniques, we split the error between the exact solution and the numerical solution into two parts which are called the temporal error and the spatial error. First, we analyze the temporal error by introducing a time-discrete system and derive some regularity of the solution of the time-discrete system. Second, by the above achievements, we obtain the τ-independent spatial error and the boundedness of the numerical solution in L∞-norm. Then, the optimal L2 and H1 error estimates for r-th order FEMs (r=1,2) are derived without any restriction on the time step size. Numerical results in both two and three dimensional spaces are presented to confirm the theoretical predictions and demonstrate the efficiency of the methods.

A plane wave least squares method for the Maxwell equations in anisotropic media

In this paper, we first consider the time-harmonic Maxwell equations with Dirichlet boundary conditions in three-dimensional anisotropic media, where the coefficients of the equations are general symmetric positive definite matrices. By using scaling transformations and coordinate transformations, we build the desired stability estimates between the original electric field and the transformed nonphysical field on the condition number of the anisotropic coefficient matrix. More importantly, we prove that the resulting approximate solutions generated by plane wave least squares (PWLS) methods have the nearly optimal L2 error estimates with respect to the condition number of the coefficient matrix. Finally, numerical results verify the validity of the theoretical results, and the comparisons between the proposed PWLS method and the existing PWDG method are also provided.

Local RBF-based penalized least-squares approximation on the sphere with noisy scattered data

In this paper we derive local L2 error estimates for penalized least-squares approximation on the d-dimensional unit sphere Sd⊆Rd+1, given noisy, scattered, local data representing an underlying function from a Sobolev space of order s>d∕2 defined on a non-empty connected open set Ω⊆Sd with Lipschitz-continuous boundary. The quadratic regularization functional has two terms, one measuring the squared pointwise ℓ2-discrepancy from the local data, the other containing the squared native space norm of a radial basis function (RBF), multiplied by a regularization parameter. The RBF is chosen so that its native space is equivalent to the (global) Sobolev space of order s on Sd. While both the data and the approximated function are local, we minimize the quadratic functional over all functions in the native space of the RBF, and obtain as exact minimizer a (global) radial basis function approximation. By choosing the RBF to be a Wendland function the resulting linear system has a sparse matrix which is easily computed. We consider three different strategies for choosing the smoothing parameter, namely Morozov’s discrepancy principle and two a priori strategies, and derive L2(Ω) error estimates for each strategy. As auxiliary tools for proving the local L2 error estimates we develop both a local L2 sampling inequality and a suitable Sobolev extension theorem. The paper concludes with numerical experiments.

Unconditional optimal error estimates of linearized backward Euler Galerkin FEMs for nonlinear Schrödinger-Helmholtz equations

In this paper, we establish unconditionally optimal error estimates for linearized backward Euler Galerkin finite element methods (FEMs) applied to nonlinear Schrodinger-Helmholtz equations. By using the temporal-spatial error splitting techniques, we split the error between the exact solution and the numerical solution into two parts which are called the temporal error and the spatial error. First, by introducing a time-discrete system, we prove the uniform boundedness for the solution of this time-discrete system in some strong norms and derive error estimates in temporal direction. Second, by the above achievements, we obtain the boundedness of the numerical solution in $L^{\infty }$ -norm. Then, the optimal L2 error estimates for r-order FEMs are derived without any restriction on the time step size. Numerical results in both two- and three-dimensional spaces are provided to illustrate the theoretical predictions and demonstrate the efficiency of the methods.

A new staggered DG method for the Brinkman problem robust in the Darcy and Stokes limits

In this paper we propose a novel staggered discontinuous Galerkin method for the Brinkman problem on general polygonal meshes. The proposed method is robust in the Stokes and Darcy limits, in addition, hanging nodes can be automatically incorporated in the construction of the method, which are desirable features in practical applications. There are three unknowns involved in our formulation, namely velocity gradient, velocity and pressure. Unlike the original staggered DG formulation proposed for the Stokes equations in Kim et al. (2013), we relax the tangential continuity of velocity and enforce different staggered continuity properties for the three unknowns, which is tailored to yield an optimal L2 error estimates for velocity gradient, velocity and pressure independent of the viscosity coefficient. Moreover, by choosing suitable projection, superconvergence can be proved for L2 error of velocity. Finally, several numerical results illustrating the good performances of the proposed method and confirming the theoretical findings are presented.

The weak Galerkin finite element method for the transport-reaction equation

We present and analyze a weak Galerkin finite element method for solving the transport-reaction equation in d space dimensions. This method is highly flexible by allowing the use of discontinuous finite element on general meshes consisting of arbitrary polygon/polyhedra. We derive the L2-error estimate of O(hk+12)-order for the discrete solution when the kth-order polynomials are used for k≥0. Moreover, for a special class of meshes, we also obtain the optimal error estimate of O(hk+1)-order in the L2-norm. A derivative recovery formula is presented to approximate the convection directional derivative and the corresponding superconvergence estimate is given. Numerical examples on compatible and non-compatible meshes are provided to show the effectiveness of this weak Galerkin method.

A linearly implicit structure-preserving Fourier pseudo-spectral scheme for the damped nonlinear Schrödinger equation in three dimensions

In this paper, we propose a linearly implicit Fourier pseudo-spectral scheme, which preserves the total mass and energy conservation laws for the damped nonlinear Schrodinger equation in three dimensions. With the aid of the semi-norm equivalence between the Fourier pseudo-spectral method and the finite difference method, an optimal L2-error estimate for the proposed method without any restriction on the grid ratio is established by analyzing the real and imaginary parts of the error function. Numerical results are addressed to confirm our theoretical analysis.

Nonconforming finite element method for a generalized nonlinear Schrödinger equation

In this paper, an efficient nonconforming finite element method (FEM) is studied with EQ1rot element for a generalized nonlinear Schrödinger equation. First, we prove a novel result of the consistency error estimate with order O(h2) for EQ1rot element which leads to the superconvergent error estimate in broken H1-norm for semi-discrete scheme, while previous work only derive convergent results for this element. Second, a linearized backward Euler scheme is established and a time-discrete system is introduced to split the error into two parts, the temporal error and the spatial error. By using a rigorous analysis for the regularity of the time-discrete system and the proved characters of EQ1rot element, the optimal L2-error estimate is obtained without any time-step restrictions, which leads to the numerical solution can be bounded in L∞-norm by an inverse inequality unconditionally. Then, the supercloseness estimate is arrived at with the above achievements. Third, global superconvergence results are deduced through interpolated postprocessing technique. At last, numerical examples are provided to confirm the theoretical analysis. Here, h is the subdivision parameter, and τ is the time step.

Improved L2 and H1 error estimates for the Hessian discretization method

AbstractThe Hessian discretization method (HDM) for fourth‐order linear elliptic equations provides a unified convergence analysis framework based on three properties namely coercivity, consistency, and limit‐conformity. Some examples that fit in this approach include conforming and nonconforming finite element methods (ncFEMs), finite volume methods (FVMs) and methods based on gradient recovery operators. A generic error estimate has been established in L2, H1, and H2‐like norms in literature. In this paper, we establish improved L2 and H1 error estimates in the framework of HDM and illustrate it on various schemes. Since an improved L2 estimate is not expected in general for FVM, a modified FVM is designed by changing the quadrature of the source term and a superconvergence result is proved for this modified FVM. In addition to the Adini ncFEM, in this paper, we show that the Morley ncFEM is an example of HDM. Numerical results that justify the theoretical results are also presented.

A [formula omitted]-finite volume element scheme for anisotropic diffusion problems on general convex quadrilateral mesh

In this paper, we study a Q1-finite volume element scheme for anisotropic diffusion problems on general convex quadrilateral mesh. It is known that the coercivity is the basement for some other theoretical results (stability, H1 and L2 error estimates, etc.) and the existing results were mainly obtained on h1+γ-parallelogram meshes with scalar diffusion coefficients. For the cases of full diffusion tensors and arbitrary convex quadrilateral meshes, we obtain a necessary and sufficient condition for the positive definiteness of the cell matrix related to the cell bilinear form. Based on this result, a sufficient condition is suggested to guarantee the coercivity of the scheme. More interesting is that, this sufficient condition covers the traditional h1+γ-parallelogram mesh assumption and has an explicit expression, by which one can easily judge on any diffusion tensor and any mesh with arbitrary mesh size h>0. Moreover, an H1 error estimate is obtained without the h1+γ-parallelogram assumption, and some numerical results are also provided to validate the theoretical results.

A modified characteristics projection finite element method for unsteady incompressible Magnetohydrodynamics equations

This paper provides a modified characteristics projection finite element method for the unsteady incompressible magnetohydrodynamics(MHD) equations. In this method, modified characteristics finite element method and the projection method will be combined for solving the unsteady incompressible MHD equations. Both the stability and the optimal error estimates both in L2 and H1 norms for the modified characteristics projection finite element method will be shown. In order to demonstrate the effectiveness of our method, we will present some numerical results at the end.

A two-grid MMOC finite element method for nonlinear variable-order time-fractional mobile/immobile advection–diffusion equations

A fully discrete two-grid modified method of characteristics (MMOC) scheme is proposed for nonlinear variable-order time-fractional advection–diffusion equations in two space dimensions. The MMOC is used to handle the advection-dominated transport and the two-grid method is designed for efficiently solving the resulting nonlinear system. Optimal L2 error estimates are derived for both the MMOC scheme and the corresponding two-grid MMOC scheme. Numerical experiments are presented to demonstrate the accuracy and the efficiency of the proposed method.

A second order numerical scheme for nonlinear Maxwell’s equations using conforming finite element

In this paper, we study and analyze a second order numerical scheme for Maxwell’s equations with nonlinear conductivity, using the Nédelec Finite Element Method (FEM). A purely explicit treatment of the nonlinear term greatly simplifies the computational effort, since we only need to solve a constant-coefficient linear system at each time step. The curl-conforming nature of the Nédelec element assures its divergence-free property. In turn, we present the linearized stability analysis for the numerical error function to obtain an optimal L2 error estimate. In more details, an O(τ2+hs) error estimate in the L2 norm yields the maximum norm bound of the numerical solution, so that the convergence analysis could be carried out at the next time step. A few numerical examples in the transverse electric (TE) case in two dimensional spaces are also presented, which demonstrate the efficiency and accuracy of the proposed numerical scheme.

A stabilized semi-implicit Fourier spectral method for nonlinear space-fractional reaction-diffusion equations

The reaction-diffusion model can generate a wide variety of spatial patterns, which has been widely applied in chemistry, biology, and physics, even used to explain self-regulated pattern formation in the developing animal embryo. In this work, a second-order stabilized semi-implicit time-stepping Fourier spectral method for the reaction-diffusion systems of equations with space described by the fractional Laplacian is developed. We adopt the temporal-spatial error splitting argument to illustrate that the proposed method is stable without imposing the CFL condition, and an optimal L2-error estimate in space is proved. We also analyze the linear stability of the stabilized semi-implicit method and obtain a practical criterion to choose the time step size to guarantee the stability of the semi-implicit method. Our approach is illustrated by solving several problems of practical interest, including the fractional Allen-Cahn, Gray-Scott and FitzHugh-Nagumo models, together with an analysis of the properties of these systems in terms of the fractional power of the underlying Laplacian operator, which are quite different from the patterns of the corresponding integer-order model.