L2 Error Estimates Research Articles

One‐level and two‐level finite element methods for the Boussinesq problem

Previous works on the convergence of numerical methods for the Boussinesq problem were conducted, while the optimal L2‐norm error estimates for the velocity and temperature are still lacked. In this paper, the backward Euler scheme is used to discrete the time terms, standard Galerkin finite element method is adopted to approximate the variables. The MINI element is used to approximate the velocity and pressure, the temperature field is simulated by the linear polynomial. Under some restriction on the time step, we firstly present the optimal L2 error estimates of approximate solutions. Secondly, two‐level method based on Stokes iteration for the Boussinesq problem is developed and the corresponding convergence results are presented. By this method, the original problem is decoupled into two small linear subproblems. Compared with the standard Galerkin method, the two‐level method not only keeps good accuracy but also saves a lot of computational cost. Finally, some numerical examples are provided to support the established theoretical analysis.

A superconvergent local discontinuous Galerkin method for nonlinear two-point boundary-value problems

In this paper, we present and analyze a superconvergent and high order accurate local discontinuous Galerkin (LDG) method for nonlinear two-point boundary-value problems (BVPs) of the form u ″ = f (t, u), which arise in a wide variety of engineering applications. We prove the L 2 stability of the LDG scheme and optimal L 2 error estimates for the solution and for the auxiliary variable that approximates the first-order derivative. The order of convergence is proved to be p + 1, when piecewise polynomials of degree at most p are used. Our numerical experiments demonstrate optimal rates of convergence. Moreover, we show that the derivatives of the LDG solutions are superconvergent with order p + 1 toward the derivatives of Gausss-Radau projections of the exact solutions. Finally, we prove that the LDG solutions are superconvergent with order p + 3/2 toward Gauss-Radau projections of the exact solutions. Our computational results indicate that the observed numerical superconvergence rate is p + 2. Our proofs are valid for arbitrary regular meshes using piecewise polynomials of degree p ≥ 1 and for the periodic, Dirichlet, and mixed boundary conditions. All proofs are valid under the hypotheses of the existence and uniqueness theorem for BVPs. Several numerical results are presented to validate the theoretical results.

Analysis of the edge finite element approximation of the Maxwell equations with low regularity solutions

We derive H(curl)-error estimates and improved L2-error estimates for the Maxwell equations approximated using edge finite elements. These estimates only invoke the expected regularity pickup of the exact solution in the scale of the Sobolev spaces, which is typically lower than 12 and can be arbitrarily close to 0 when the material properties are heterogeneous. The key tools for the analysis are commuting quasi-interpolation operators in H(curl)- and H(div)-conforming finite element spaces and, most crucially, newly-devised quasi-interpolation operators delivering optimal estimates on the decay rate of the best-approximation error for functions with Sobolev smoothness index arbitrarily close to 0. The proposed analysis entirely bypasses the technique known in the literature as the discrete compactness argument.

Weak Galerkin mixed finite element method for heat equation

In this paper, we apply a new weak Galerkin mixed finite element method (WGMFEM) with stabilization term to solve heat equations. This method allows the usage of totally discontinuous functions in the approximation space. The WGMFEM is capable of providing very accurate numerical approximations for both the primary and flux variables. In addition, we develop and analyze the error estimates for both continuous and discontinuous time WGMFEM schemes. Optimal order error estimates in both L2 and triple-bar ⫼⋅⫼ norms are established, respectively. Finally, numerical tests are conducted to illustrate the theoretical results.

Unconditional optimal error estimates for BDF2-FEM for a nonlinear Schrödinger equation

This paper analyzes unconditional optimal error estimates for a 2-step backward differentiation formula (BDF2) method for a nonlinear Schrödinger equation. In the analysis, we split an error estimate into two parts, one from the spatial discretization and the other from the temporal discretization. We present the boundedness of the solution of the time-discrete system in the certain strong norms, and the error estimates for time discretization. By these boundedness and temporal error estimates, we obtain the L2 error estimates without any conditions on a time step size. Numerical experiments are provided to validate our analysis and check the efficiency of our method.

Symmetric interior penalty Galerkin approaches for two-dimensional parabolic interface problems with low regularity solutions

This work presents novel finite element approaches for solving a parabolic partial differential equation with discontinuous coefficients and low regularity solutions in a bounded convex polyhedral domain. A spatial semi-discretization based on symmetric interior penalty Galerkin (SIPG) approximations is constructed and analyzed by using discontinuous piecewise linear functions. For smooth initial data, spatial errors in the broken L2, H1 and L2(H1) norms are proven to be optimal with respect to low regularity solutions, which are only piecewise H1+s smooth with 0<s≤1. Furthermore we present the full SIPG discretizations based on an Euler backward finite difference time discretization. The proposed approximations are shown to be unconditionally stable, and have nearly the optimal L2(L2) and L2(H1) error estimates, even when the regularities of the solutions are low on the whole domain. The error estimates are optimal with respect to time, sharply depending on the indexes in the global and local regularity. Numerical experiments for two-dimensional parabolic interface problems verify the theoretical convergence rates.

A posteriori error estimates and adaptivity for the discontinuous Galerkin solutions of nonlinear second-order initial-value problems

In this paper, we propose and analyze an efficient and reliable a posteriori error estimator of residual-type for the discontinuous Galerkin (DG) method applied to nonlinear second-order initial-value problems for ordinary differential equations. This estimator is simple, efficient, and asymptotically exact. We use our recent optimal L2 error estimates and superconvergence results of Baccouch [15] to show that the significant parts of the DG discretization errors are proportional to the (p+1)-degree right Radau polynomial, when polynomials of total degree not exceeding p are used. These new results allow us to construct a residual-based a posteriori error estimator which is obtained by solving a local residual problem with no initial condition on each element. We prove that, for smooth solutions, the proposed a posteriori error estimator converges to the actual error in the L2-norm with order of convergence p+2. Computational results indicate that the theoretical order of convergence is sharp. By adding the a posteriori error estimate to the DG solution, we obtain a post-processed approximation which superconverges with order p+2 in the L2-norm. Moreover, we demonstrate the effectiveness of the this error estimator. Finally, we present a local adaptive mesh refinement (AMR) procedure that makes use of our local a posteriori error estimate. Our proofs are valid for arbitrary regular meshes and for Pp polynomials with p≥1. Several numerical results are presented to validate the theoretical results.

Unconditional optimal error estimates of a two-grid method for semilinear parabolic equation

In this paper, the error analysis of a two-grid method (TGM) with backward Euler scheme is discussed for semilinear parabolic equation. Contrary to the conventional finite element analysis, the error between exact solution and finite element solution is split into two parts (temporal error and spatial error) by introducing a corresponding time-discrete system. This can lead to the spatial error independent of τ (time step). Secondly, based on the above technique, optimal error estimates in L2 and H1-norms of TGM solution are deduced unconditionally, while previous works always require a certain time step size condition. Finally, a numerical experiment is provided to confirm the theoretical analysis.

Error analysis of a compact finite difference method for fourth-order nonlinear elliptic boundary value problems

This paper is concerned with a compact finite difference method with non-isotropic mesh sizes for a two-dimensional fourth-order nonlinear elliptic boundary value problem. By the discrete energy analysis, the optimal error estimates in the discrete L2, H1 and L∞ norms are obtained without any constraint on the mesh sizes. The error estimates show that the compact finite difference method converges with the convergence rate of fourth-order. Based on a high-order approximation of the solution, a Richardson extrapolation algorithm is developed to make the final computed solution sixth-order accurate. Numerical results demonstrate the high-order accuracy of the compact finite difference method and its extrapolation algorithm in the discrete L2, H1 and L∞ norms.

Optimal error analysis of Crank–Nicolson schemes for a coupled nonlinear Schrödinger system in 3D

The paper is concerned with the time step condition of the commonly-used semi-implicit Crank–Nicolson finite difference schemes for a coupled nonlinear Schrödinger system in three dimensional space. We present the optimal L2 error estimate without any restriction on time step, while all previous works require certain time step conditions. Our approach is based on a rigorous analysis in both real and imaginary parts of the energy estimate (inequality) of the error function. Numerical examples for both two-dimensional and three-dimensional models are investigated and numerical results illustrate our theoretical analysis.

The approximation of parabolic equations involving fractional powers of elliptic operators

We study the numerical approximation of a time dependent equation involving fractional powers of an elliptic operator L defined to be the unbounded operator associated with a Hermitian, coercive and bounded sesquilinear form on H01(Ω). The time dependent solution u(x,t) is represented as a Dunford–Taylor integral along a contour in the complex plane.The contour integrals are approximated using sinc quadratures. In the case of homogeneous right-hand-sides and initial value v, the approximation results in a linear combination of functions (zqI−L)−1v∈H01(Ω) for a finite number of quadrature points zq lying along the contour. In turn, these quantities are approximated using complex valued continuous piecewise linear finite elements.Our main result provides L2(Ω) error estimates between the solution u(⋅,t) and its final approximation. Numerical results illustrating the behavior of the algorithms are provided.

Second-order approximation scheme combined with [formula omitted]-Galerkin MFE method for nonlinear time fractional convection–diffusion equation

In this article, a second-order approximation scheme combined with an H1-Galerkin mixed finite element (MFE) method for solving nonlinear convection–diffusion equation with time fractional derivative is proposed and analyzed. By introducing an auxiliary variable, a coupled system is formulated, then the spatial direction is approximated by H1-Galerkin MFE method and the temporal fractional derivative and integer derivative are discretized by second-order weighted and shifted Grünwald difference (WSGD) formula and linearized second-order difference scheme, respectively. The optimal priori error estimates in L2 and H1-norm for the unknown function and the auxiliary variable with second-order convergent rate in time are obtained. Compared to the commonly used L1-approximation with (2−α)th-order convergence rate, our method can arrive at the order 2 in time. What is more, compared with the standard finite element method, our method can well approximate the auxiliary variable. Finally, the detailed computational process of the studied numerical algorithm is shown and a nonlinear numerical example with calculated data and some figures is provided to verify our theoretical analysis.

ANISOTROPIC QUASI-WILSON ELEMENT WITH CONFORMING FINITE ELEMENT APPROXIMATION FOR COUPLED CONTINUUM PIPE-FLOW/DARCY MODEL IN KARST AQUIFERS

This paper presents a numerical method for solving systems of partial differential equations describing flow in porous media with an embedded and inclined conduit pipe. This work considers a coupled continuum pipe-flow/Darcy model. The numerical schemes presented are based on combinations of the quasi-Wilson element on anisotropic mesh and the conforming finite element on regular mesh. The existence and uniqueness of the approximation solution are obtained. Optimal error estimates in both L2 and H1 norms are obtained independent of the regularity condition on the mesh. Numerical examples show the accuracy and efficiency of the proposed scheme.

A short note on the error estimates of Yuan–Shu discontinuous Galerkin method based on non-polynomial approximation spaces

In the article [Yuan and Shu (2006) [17]], Yuan and Shu have developed discontinuous Galerkin (DG) methods based on non-polynomial approximation spaces for solving time dependent problems. The authors established L2 error estimates of the proposed methods and also presented a criterion for the choice of basis functions of the non-polynomial spaces to have the same approximation rates as those of polynomial finite element spaces of the same dimension. However, the verification that some of approximation spaces do satisfy the criterion has been performed only to limited types of approximation spaces. In this regards, the aim of this short note is to fill the gap. We prove that the criterion of Yuan and Shu can be satisfied by a wide class of basis functions, including the well-known basis functions such as trigonometric and exponential polynomials.

On a Legendre Tau Method for Fractional Boundary Value Problems with a Caputo Derivative

Abstract In this paper, we revisit a Legendre-tau method for two-point boundary value problems with a Caputo fractional derivative in the leading term, and establish an L 2 error estimate for smooth solutions. Further, we apply the method to the Sturm-Liouville problem. Numerical experiments indicatethat for the source problem, it converges steadily at an algebraic rate even for nonsmooth data, and the convergence rate enhances with problem data regularity, whereas for the Sturm-Liouville problem, it always yields excellent convergence for eigenvalue approximations.

Accuracy of Finite Element Methods for Boundary-Value Problems of Steady-State Fractional Diffusion Equations

Optimal-order error estimates in the energy norm and the $$L^2$$L2 norm were previously proved in the literature for finite element methods of Dirichlet boundary-value problems of steady-state fractional diffusion equations under the assumption that the true solutions have desired regularity and that the solution to the dual problem has full regularity for each right-hand side. We show that the solution to the homogeneous Dirichlet boundary-value problem of a one-dimensional steady-state fractional diffusion equation of constant coefficient and source term is not necessarily in the Sobolev space $$H^1$$H1. This fact has the following implications: (i) Up to now, there are no verifiable conditions on the coefficients and source terms of fractional diffusion equations in the literature to ensure the high regularity of the true solutions, which are in turn needed to guarantee the high-order convergence rates of their numerical approximations. (ii) Any Nitsche-lifting based proof of optimal-order $$L^2$$L2 error estimates of finite element methods in the literature is invalid. We present numerical results to show that high-order finite element methods for a steady-state fractional diffusion equation with smooth data and source term fail to achieve high-order convergence rates. We present a preliminary development of an indirect finite element method, which reduces the solution of fractional diffusion equations to that of second-order diffusion equations postprocessed by a fractional differentiation. We prove that the corresponding high-order methods achieve high-order convergence rates even though the true solutions are not smooth, provided that the coefficient and source term of the problem have desired regularities. Numerical experiments are presented to substantiate the theoretical estimates.

A Nonintrusive Stratified Resampler for Regression Monte Carlo: Application to Solving Nonlinear Equations

Our goal is to solve certain dynamic programming equations associated to a given Markov chain X, using a regression-based Monte Carlo algorithm. More specifically, we assume that the model for X is not known in full detail and only a root sample X1, . . . , XM of such process is available. By a stratification of the space and a suitable choice of a probability measure ν, we design a new resampling scheme that allows to compute local regressions (on basis functions) in each stratum. The combination of the stratification and the resampling allows to compute the solution to the dynamic programming equation (possibly in large dimensions) using only a relatively small set of root paths. To assess the accuracy of the algorithm, we establish non-asymptotic error estimates in L2(ν). Our numerical experiments illustrate the good performance, even with M = 20 − 40 root paths.

A second-order decoupled implicit/explicit method of the 3D primitive equations of ocean II: finite element spatial discretization

Summary A fully discrete second-order decoupled implicit/explicit method is proposed for solving 3D primitive equations of ocean in the case of Dirichlet boundary conditions on the side, where a second-order decoupled implicit/explicit scheme is used for time discretization, and a finite element method based on the P1(P1) − P1−P1(P1) elements for velocity, pressure and density is used for spatial discretization of these primitive equations. Optimal H1−L2−H1 error estimates for numerical solution uhn,phn,θhn and an optimal L2 error estimate for uhn,θhn are established under the convergence condition of 0 < h≤β1,0 < τ≤β2, and τ≤β3h for some positive constants β1,β2, and β3. Furthermore, numerical computations show that the H1−L2−H1 convergence rate for numerical solution uhn,phn,θhn is of O(h + τ2) and an L2 convergence rate for uhn,θhn is O(h2+τ2) with the assumed convergence condition, where h is a mesh size and τ is a time step size. More practical calculations are performed as a further validation of the numerical method. Copyright © 2016 John Wiley & Sons, Ltd.

Error estimates for approximations of distributed order time fractional diffusion with nonsmooth data

Abstract In this work, we consider the numerical solution of a distributed order subdiffusion model, arising in the modeling of ultra-slow diffusion processes. We develop a space semidiscrete scheme based on the Galerkin finite element method, and establish error estimates optimal with respect to data regularity in L 2(Ω) and H 1(Ω) norms for both smooth and nonsmooth initial data. Further, we propose two fully discrete schemes, based on the Laplace transform and convolution quadrature generated by the backward Euler method, respectively, and provide optimal L 2(Ω) error estimates, which exhibits exponential convergence and first-order convergence in time, respectively. Extensive numerical experiments are provided to verify the error estimates for both smooth and nonsmooth initial data.

Efficient expanded mixed finite element method for the quasilinear elliptic problems

With the rapid development of computing power, advanced modelling and numerical simulation technology has become a useful and powerful tool. Some efficient numerical methods are presented. In this article, expanded mixed finite element method is introduced to solve the quasilinear elliptic problems. This method expands the traditional mixed finite element method in the sense that three variables are explicitly treated simultaneously. Existence and uniqueness of the discrete approximation are demonstrated. L2-error estimates for three variables are presented. Numerical examples are carried out to validate the theoretical analysis.