Optimal Order Error Estimates Research Articles

A locking-free finite difference method on staggered grids for linear elasticity problems

A finite difference method on staggered grids is constructed on general nonuniform rectangular partition for linear elasticity problems. Stability, optimal-order error estimates in discrete H1-norms on general nonuniform grids and second-order superconvergence on almost uniform grids have been obtained. These theoretical results are uniform about the Lamé constant λ∈(0,∞) so the finite difference method is locking-free. The method and theoretical results can be extended to three dimensional problems. Numerical experiments using the method show agreement of the numerical results with theoretical analysis.

A new weak Galerkin finite element scheme for general second-order elliptic problems

A new weak Galerkin (WG) finite element scheme is presented for general second-order elliptic problems in this paper. In this new scheme, a skew symmetric form has been used for handling the convection term. The advantage of the new scheme is that the system of linear equations from the scheme is positive definite and one might easily get the well-posedness of the system. In this scheme, the WG elements are designed to have the form of ( P k ( T ) , P k − 1 ( e ) ) . That is, we choose the polynomials of degree k ≥ 1 on each element and the polynomials of degree k − 1 on the edge/face of each element. As a result, fewer degrees of freedom are generated in the new WG finite element scheme. It is also worth pointing out that the WG finite element scheme is established on finite element partitions consisting of arbitrary shape of polygons/polyhedra which are shape regular. Optimal-order error estimates are presented for the corresponding numerical approximation in various norms. Some numerical results are reported to confirm the theory.

Optimal Order Error Estimates for Discontinuous Galerkin Methods for the Wave Equation

In this paper, we derive optimal order error estimates for spatially semi-discrete and fully discrete schemes to numerically solve the second-order wave equation. The numerical schemes are constructed with the discontinuous Galerkin (DG) discretization for the spatial variable and the centered second-order finite difference approximation for the temporal variable. Under appropriate regularity assumptions on the solution, the schemes are shown to enjoy the optimal order error bounds in terms of both the spatial mesh-size and the time-step. In Grote and Schotzau (J Sci Comput 40:257–272, 2009), a fully discrete DG scheme is studied with an explicit finite difference temporal discretization where a CFL condition is required on the mesh-size and the time-step, and optimal order error estimates are derived in the $$L^2(\Omega )$$ -norm. In comparison, for our fully discrete DG schemes, we do not require a CFL condition on the mesh-size and the time-step, and our optimal order error estimates are derived for the $$H^1(\Omega )$$ -like norm and the $$L^2(\Omega )$$ norm. Numerical simulation results are reported to illustrate theoretically predicted convergence orders in the $$H^1(\Omega )$$ and $$L^2(\Omega )$$ norms.

Numerical analysis of the energy-dependent radiative transfer equation

Abstract The energy-dependent form of the radiative transfer equation (RTE) is important in a variety of applications. In a previous paper the well-posedness and an energy discretization for the energy-dependent RTE were investigated. In this paper the fully discrete scheme is introduced and analysed. Optimal-order error estimates and a well-posedness analysis of the discrete system are provided. The theoretical results are validated through numerical examples.

A Modified Characteristics-mixed Finite Element for Semiconductor Device of Heat Conduction and Numerical Analysis

Numerical simulation of a three-dimensional semiconductor device of heat conduction is a fundamental problem in modern information science. The mathematical model is formulated by a nonlinear system of initial-boundary problem, which is interpreted by four partial differential equations: an elliptic equation for electrostatic potential, two convection-diffusion equations for electron concentration and hole concentration, a heat conduction equation for temperature. The electrostatic potential appears within the latter three equations, and the electric field strength controls the concentrations and the temperature. The electric field potential is solved by a mixed finite element method, and the electric field strength is obtained simultaneously. The first order of the accuracy is improved for the latter. The concentrations and temperature are computed by the characteristics-finite element method, where the characteristic approximation is adopted for the hyperbolic term and finite element method is use to treat the diffusion. The composite computational scheme can solve the convection-dominated diffusion equations well because it can cancel numerical dispersion and nonphysical oscillation. The temperature is computed by finite element method, and an interesting simulation tool is proposed for solving semiconductor device problem numerically. By using the technique of a priori estimates of differential equations, an optimal order error estimates is obtained. A theoretical work is shown for numerical simulation of information science, and the actual problem is solved well.

New discretization schemes for time-harmonic Maxwell equations by weak Galerkin finite element methods

Abstract This paper introduces new discretization schemes for time-harmonic Maxwell equations in a connected domain by using the weak Galerkin (WG) finite element method. The corresponding WG algorithms are analyzed for their stability and convergence. Error estimates of optimal order in various discrete Sobolev norms are established for the resulting finite element approximations.

A regularization framework for mildly ill-posed problems connected with pseudo-differential operator

Recently filter-based regularization methods have been well investigated for ill-posed problems when the forward operators are compact. There are many ill-posed problems connected with pseudo-differential operators. But there is no uniform method for this kind of problems. The work on generalization of filter-based regularization methods to pseudo-differential operator is necessary. In this paper, we present a regularization framework for solving the mildly ill-posed problems involved pseudo-differential operators. A general regularization method for this kind of problems is given. The order-optimal error estimates are derived under the usual source conditions. As an example, a new fractional Tikhonov regularization method could be cast into the general framework. Numerical experiments are conducted for showing the validity of the new fractional Tikhonov method.

A discontinuous least-squares finite-element method for second-order elliptic equations

ABSTRACTIn this paper, a discontinuous least-squares (DLS) finite-element method is introduced. The novelty of this work is twofold, to develop a DLS formulation that works for general polytopal meshes and to provide rigorous error analysis for it. This new method provides accurate approximations for both the primal and the flux variables. We obtain optimal-order error estimates for both the primal and the flux variables. Numerical examples are tested for polynomials up to degree 4 on non-triangular meshes, i.e. on rectangular and hexagonal meshes.

Hybridized Discontinuous Galerkin Method for Elliptic Interface Problems: Error Estimates Under Low Regularity Assumptions of Solutions

New hybridized discontinuous Galerkin (HDG) methods for the interface problem for elliptic equations are proposed. Unknown functions of our schemes are $u_h$ in elements and $\hat{u}_h$ on inter-element edges. That is, we formulate our schemes without introducing the flux variable. Our schemes naturally satisfy the Galerkin orthogonality. The solution $u$ of the interface problem under consideration may not have a sufficient regularity, say $u|_{\Omega_1}\in H^2(\Omega_1)$ and $u|_{\Omega_2}\in H^2(\Omega_2)$, where $\Omega_1$ and $\Omega_2$ are subdomains of the whole domain $\Omega$ and $\Gamma=\partial\Omega_1\cap\partial\Omega_2$ implies the interface. We study the convergence, assuming $u|_{\Omega_1}\in H^{1+s}(\Omega_1)$ and $u|_{\Omega_2}\in H^{1+s}(\Omega_2)$ for some $s\in (1/2,1]$, where $H^{1+s}$ denotes the fractional order Sobolev space. Consequently, we succeed in deriving optimal order error estimates in an HDG norm and the $L^2$ norm. Numerical examples to validate our results are also presented.

An expanded mixed finite element simulation for two-sided time-dependent fractional diffusion problem

In this paper, we consider a time-dependent diffusion problem with two-sided Riemann-Liouville fractional derivatives. By introducing a fractional-order flux as auxiliary variable, we establish the saddle-point variational formulation, based on which we employ a locally conservative mixed finite element method to approximate the unknown function, its derivative and the fractional flux in space and use the backward Euler scheme to discrete the time derivative, and thus propose a fully discrete expanded mixed finite element procedure. We prove the well-posedness and the optimal order error estimates of the proposed procedure for a sufficiently smooth solution. Numerical experiments are presented to confirm our theoretical findings.

Weak Galerkin Finite Element Method for the Unsteady Stokes Equation

The Weak Galerkin (WG) finite element method for the unsteady Stokes equations in the primary velocity-pressure formulation is introduced in this paper. Optimal-order error estimates are established for the corresponding numerical approximation in an H1 norm for the velocity, and L2 norm for both the velocity and the pressure by use of the Stokes projection.

A decoupled Crank-Nicolson time-stepping scheme for thermally coupled magneto-hydrodynamic system

Thermally coupled magneto-hydrodynamics (MHD) studies the dynamics of electro-magnetically and thermally driven flows,involving MHD equations coupled with heat equation. We introduce a partitioned method that allows one to decouplethe MHD equations from the heat equation at each time step and solve them separately. The extrapolated Crank-Nicolson time-stepping scheme is used for time discretizationwhile mixed finite element method is used for spatial discretization. We derive optimal order error estimates in suitable norms without assuming any stability condition or restrictions on the time step size. We prove the unconditional stability of the scheme. Numerical experiments are used to illustrate the theoretical results.

Lavrentiev's regularization method for nonlinear ill-posed equations in Banach spaces

In this paper, we deal with nonlinear ill-posed problems involving m-accretive mappings in Banach spaces. We consider a derivative and inverse free method for the implementation of Lavrentiev regularization method. Using general Hölder type source condition we obtain an optimal order error estimate. Also we consider the adaptive parameter choice strategy proposed by Pereverzev and Schock (2005) for choosing the regularization parameter.

Error estimates for the simplified iteratively regularized Gauss–Newton method in Banach spaces under a Morozov-type stopping rule

Abstract Jin Qinian and Min Zhong [10] considered an iteratively regularized Gauss–Newton method in Banach spaces to find a stable approximate solution of the nonlinear ill-posed operator equation. They have considered a Morozov-type stopping rule (Rule 1) as one of the criterion to stop the iterations and studied the convergence analysis of the method. However, no error estimates have been obtained for this case. In this paper, we consider a modified variant of the method, namely, the simplified Gauss–Newton method under both an a priori as well as a Morozov-type stopping rule. In both cases, we obtain order optimal error estimates under Hölder-type approximate source conditions. An example of a parameter identification problem for which the method can be implemented is discussed in the paper.

Numerical Analysis for the Pure Neumann Control Problem Using the Gradient Discretisation Method

Abstract The article discusses the gradient discretisation method (GDM) for distributed optimal control problems governed by diffusion equation with pure Neumann boundary condition. Using the GDM framework enables to develop an analysis that directly applies to a wide range of numerical schemes, from conforming and non-conforming finite elements, to mixed finite elements, to finite volumes and mimetic finite differences methods. Optimal order error estimates for state, adjoint and control variables for low-order schemes are derived under standard regularity assumptions. A novel projection relation between the optimal control and the adjoint variable allows the proof of a super-convergence result for post-processed control. Numerical experiments performed using a modified active set strategy algorithm for conforming, non-conforming and mimetic finite difference methods confirm the theoretical rates of convergence.

A weak Galerkin finite element method for a coupled Stokes–Darcy problem on general meshes

In this paper, we introduce and analyze a weak Galerkin finite element method for numerically solving the coupling of fluid flow with porous media flow. Flows are governed by the Stokes equations in primal velocity–pressure formulation and Darcy equation in the second order primary formulation, respectively, and the corresponding transmission conditions are given by mass conservation, balance of normal forces, and the Beavers–Joseph–Saffman law. By using the weak Galerkin approach, we consider the two-dimensional problem with the usual polynomials of degree k ≥ 1 for the velocity and hydraulic head, while polynomials of degree k − 1 for the pressure, the velocity and hydraulic head is enhanced by polynomials of degree k − 1 on the edge of a finite element partition. This new method has a lot of attractive computational features: more general finite element partitions of arbitrary polygons with certain shape regularity, fewer degrees of freedom and parameter free. Stability and error estimates of optimal order are obtained. Moreover, numerical experiences are presented to illustrate the good performance, confirm the optimal order of convergence and verify the efficiency of the proposed weak Galerkin method in this paper. We also deal with finite element partitions consisting of general meshes, such as triangular mesh, quadrilateral mesh, hexagonal-dominant mesh and Voronoi mesh for the numerical weak Galerkin approximation.

Discrete maximal regularity and the finite element method for parabolic equations

Maximal regularity is a fundamental concept in the theory of partial differential equations. In this paper, we establish a fully discrete version of maximal regularity for a parabolic equation. We derive various stability results in $L^p(0,T;L^q(\Omega))$ norm, $p,q\in (1,\infty)$ for the finite element approximation with the mass-lumping to the linear heat equation. Our method of analysis is an operator theoretical one using pure imaginary powers of operators and might be a discrete version of G.~Dore and A.~Venni (On the closedness of the sum of two closed operators. \emph{Math.\ Z.}, 196(2):189--201, 1987). As an application, optimal order error estimates in that norm are proved. Furthermore, we study the finite element approximation for semilinear heat equations with locally Lipschitz continuous nonlinearity and offer a new method for deriving optimal order error estimates. Some interesting auxiliary results including discrete Gagliardo-Nirenberg and Sobolev inequalities are also presented.

An analysis of the TDNNS method using natural norms

The tangential-displacement normal-normal-stress (TDNNS) method is a finite element method for mixed elasticity. As the name suggests, the tangential component of the displacement vector as well as the normal-normal component of the stress are the degrees of freedom of the finite elements. The TDNNS method was shown to converge of optimal order, and to be robust with respect to shear and volume locking. However, the method is slightly nonconforming, and an analysis with respect to the natural norms of the arising spaces was still missing. We present a sound mathematical theory of the infinite dimensional problem using the space {{mathbf {H}}}(mathbf {curl}) for the displacement. We define the space {underline{{mathbf {H}}}}({text {div}},mathbf {{div}}) for the stresses and provide trace operators for the normal-normal stress. Moreover, the finite element problem is shown to be stable with respect to the {{mathbf {H}}}(mathbf {curl}) and a discrete {underline{{mathbf {H}}}}({text {div}},mathbf {{div}}) norm. A-priori error estimates of optimal order with respect to these norms are obtained. Beside providing a new analysis for the elasticity equation, the numerical techniques developed in this paper are a foundation for more complex models from structural mechanics such as Reissner Mindlin plate equations, see Pechstein and Schöberl (Numerische Mathematik 137(3):713–740, 2017).

Simplified Generalized Gauss–Newton Method for Nonlinear Ill-Posed Operator Equations in Hilbert Scales

Abstract In this paper, we study the simplified generalized Gauss–Newton method in a Hilbert scale setting to get an approximate solution of the ill-posed operator equation of the form F ⁢ ( x ) = y {F(x)=y} where F : D ⁢ ( F ) ⊆ X → Y {F:D(F)\subseteq X\to Y} is a nonlinear operator between Hilbert spaces X and Y. Under suitable nonlinearly conditions on F, we obtain an order optimal error estimate under the Morozov type stopping rule.

A Weak Galerkin Method for the Reissner–Mindlin Plate in Primary Form

A new finite element method is developed for the Reissner–Mindlin equations in its primary form by using the weak Galerkin approach. Like other weak Galerkin finite element methods, this one is highly flexible and robust by allowing the use of discontinuous approximating functions on arbitrary shape of polygons and, at the same time, is parameter independent on its stability and convergence. Error estimates of optimal order in mesh size h are established for the corresponding weak Galerkin approximations. Numerical experiments are conducted for verifying the convergence theory, as well as suggesting some superconvergence and a uniform convergence of the method with respect to the plate thickness.