Optimal Error Estimates Research Articles

Unfitted finite element methods based on correction functions for Stokes flows with singular forces of low regularity

This paper presents an unfitted finite element method based on correction functions for solving stationary Stokes flows with singular forces acting on an immersed interface. It has been shown that the singular force is equivalent to a nonhomogeneous jump condition on the interface. In this paper, we consider the case that the jump has a low regularity so that it is impossible to use pointwise values on the interface to construct correction functions, as done in Guzmán et al. (2016). The natural way to deal with the problem is to use mean values of the jump on the parts of the interface cut by elements, instead of using pointwise values. However, we show that it may cause instability and the constant in the error estimate may depend on the interface location relative to the mesh. Inspired by Guo et al. (2019), we use a larger fictitious circle to overcome these issues. Associated with the correction functions, we consider two stable finite element pairs: the Mini element and the P2−P0 element, including the cases of continuous and discontinuous pressures. The optimal approximation capabilities of the correction functions and optimal error estimates of the finite element methods are both derived with a hidden constant independent of the interface location relative to the mesh. Numerical examples are provided to validate the theoretical results.

An iterative method based on Nesterov acceleration for identifying space-dependent source term in a time-fractional diffusion-wave equation

In this paper, we investigate an inverse problem of identifying a space-dependent source in a time-fractional diffusion-wave equation by using the final time data. Since the problem is ill-posed, we propose an iterative method based on the Nesterov acceleration strategy to deal with it. Two kinds of convergence rates for the regularized solution are given under both the a priori and a posteriori regularization parameter choice rules. Compared with the classical regularization methods, our method is easy to implement with a fewer number of iterations and can always yield the order optimal error estimates provided the iteration parameter is chosen large enough. Some numerical examples including one-dimensional and two-dimensional cases are presented to illustrate the validity and effectiveness of the proposed method.

A fully-decoupled discontinuous Galerkin method for the nematic liquid crystal flows with SAV approach

In this paper, we consider the numerical approximation of a hydrodynamic Ericksen–Leslie system, which describes the macroscopic continuum of liquid crystals. A numerical challenge for solving the governing system is how to construct a suitable numerical scheme so that it is unconditionally energy stable at the discrete level. We propose a novel linear, decoupled fully discrete scheme, which is based on the design of the scalar auxiliary variable (SAV) for the nonlinear potential and combined with the discontinuous Galerkin (DG) for spatial discretization, the pressure-projection method for the Navier–Stokes equations, and implicit–explicit (IMEX) approaches for the highly nonlinear and coupling terms. We rigorously prove the unconditional energy stability and optimal error estimates of the proposed scheme, especially the optimal error bounds for the pressure. Finally, several numerical examples are performed to numerically demonstrate the accuracy, energy stability, and applicability of the proposed scheme.

Discontinuous Galerkin approximations to elliptic and parabolic problems with a Dirac line source

The analyses of interior penalty discontinuous Galerkin methods of any order k for solving elliptic and parabolic problems with Dirac line sources are presented. For the steady state case, we prove convergence of the method by deriving a priori error estimates in the L2 norm and in weighted energy norms. In addition, we prove almost optimal local error estimates in the energy norm for any approximation order. Further, almost optimal local error estimates in the L2 norm are obtained for the case of piecewise linear approximations whereas suboptimal error bounds in the L2 norm are shown for any polynomial degree. For the time-dependent case, convergence of semi-discrete and of backward Euler fully discrete scheme is established by proving error estimates in L2 in time and in space. Numerical results for the elliptic problem are added to support the theoretical results.

Any order spectral volume methods for diffusion equations using the local discontinuous Galerkin formulation

In this paper, we present and study two spectral volume (SV) schemes of arbitrary order for diffusion equations by using the local discontinuous Galerkin formulation to discretize the viscous flux. The basic idea of the scheme is to rewrite the diffusion equation into an equivalent first-order system first, and then use the SV method to solve the system. The SV scheme is designed with control volumes constructed by using the Gauss points or Radau points in subintervals of the underlying meshes, which leads to two SV schemes referred to as LSV and RSV schemes, respectively. The stability analysis for the linear diffusion equations based on alternating fluxes are provided, and optimal error estimates are established for both the exact solution and the auxiliary variable. Furthermore, a rigorous mathematical proof are given to demonstrate that the proposed RSV method is identical to the standard LDG method when applied to constant diffusion problems. Numerical experiments are presented to demonstrate the stability, accuracy and performance of the two SV schemes for both linear and nonlinear diffusion equations.

Discontinuous Galerkin methods for stochastic Maxwell equations with multiplicative noise

In this paper we propose and analyze finite element discontinuous Galerkin methods for the one- and two-dimensional stochastic Maxwell equations with multiplicative noise. The discrete energy law of the semi-discrete DG methods were studied. Optimal error estimate of the semi-discrete method is obtained for the one-dimensional case, and the two-dimensional case on both rectangular meshes and triangular meshes under certain mesh assumptions. Strong Taylor 2.0 scheme is used as the temporal discretization. Both one- and two-dimensional numerical results are presented to validate the theoretical analysis results.

Error Estimates of High-order Conservative Schemes for the Generalized Nonlinear Schrödinger Equation in One Dimension

In this paper, we use the high-order difference operator Dμ to compensate the central difference operator . Then, we discretize time by the time-midpoint method. Fully discrete schemes are obtained for the generalized nonlinear Schrödinger equation. For the high-order methods, conservation of mass and energy can be proved theoretically in the discrete sense. Then, to establish the optimal error estimates without any requirement on the grid ratio, we adopt the cut-off function technique. The error estimates of the schemes are proved to be of O(τ2 + h2(μ+1)) with time step size τ and mesh size h. Some numerical experiments are given to verify the accuracy order. In addition, the dynamics of the generalized nonlinear Schrödinger equation in one dimension are simulated.

Optimal error estimates of a linearized second-order BDF scheme for a nonlocal parabolic problem

PurposeThis paper focuses on the unconditionally optimal error estimates of a linearized second-order scheme for a nonlocal nonlinear parabolic problem. The first step of the scheme is based on Crank–Nicholson method while the second step is the second-order BDF method.Design/methodology/approachA rigorous error analysis is done, and optimal L2 error estimates are derived using the error splitting technique. Some numerical simulations are presented to confirm the study’s theoretical analysis.FindingsOptimal L2 error estimates and energy norm.Originality/valueThe goal of this research article is to present and establish the unconditionally optimal error estimates of a linearized second-order BDF finite element scheme for the reaction-diffusion problem. An optimal error estimate for the proposed methods is derived by using the temporal-spatial error splitting techniques, which split the error between the exact solution and the numerical solution into two parts, that is, the temporal error and the spatial error. Since the spatial error is not dependent on the time step, the boundedness of the numerical solution in L∞-norm follows an inverse inequality immediately without any restriction on the grid mesh.

The virtual element method for general variational–hemivariational inequalities with applications to contact mechanics

This paper mainly analyzes the general elliptic variational–hemivariational inequalities with or without constraints by using the virtual element method. The approximations can be internal or external and a Céa’s type inequality is derived for a priori error estimates. Then, we apply the results to a variational–hemivariational inequality arising in frictional contact problems, and the optimal order error estimate is obtained for the linear virtual element solution under appropriate solution regularity assumptions. Finally, numerical simulation results are reported to show the performance of the proposed method, in particular, numerical convergence orders are in good agreement with the theoretical predictions.

Optimal a priori error estimate of relaxation-type linear finite element method for nonlinear Schrödinger equation

In this paper, we study, analyze and numerically validate a conservative relaxation-type linear finite element method (FEM) for the nonlinear Schrödinger equation. The method avoids solving the complex nonlinear system and preserves the discrete mass and energy. A key to our analysis is that the errors are split into the temporal error and the spatial error by introducing the corresponding time-discrete system. Therefore, we derive the optimal L2 and semi-H1 error estimates without any coupling condition between time step τ and space size h. Some numerical experiments not only demonstrate the method’s optimal convergence rates but also confirm the conservation laws during long time simulations.

Electric potential-robust iterative analysis of charge-conservative conforming FEM for thermally coupled inductionless MHD system

This paper mainly considers electric potential-robust iterative methods based on conservation of charge in Lipschitz domain for the 2D/3D stationary thermally coupled inductionless MHD equations. Based on the hybrid finite element method, the unknowns of hydrodynamic are discretized by the stable velocity–pressure finite element pair, and the current density along with electric potential are similarly discretized by the conforming finite element pair in H(div,Ω)×L2(Ω). It is proved especially that the optimal error estimates of velocity, current density, temperature and pressure do not depend on electric potential. And on account of the strong nonlinearity of the equations, we present three coupled iterative methods, namely, the Stokes, Newton and Oseen iterations and the convergence and stability under different uniqueness conditions are analyzed strictly. The theoretical analysis is validated by the given numerical results, and for the proposed methods, the applicability and effectiveness are demonstrated.

A high-order numerical scheme for stochastic optimal control problem

In this paper, we propose a local discontinuous Galerkin (LDG) method for fully nonlinear second-order PDEs in multiple space dimensions, which can serve as a high-order numerical scheme for stochastic optimal control problems. The optimal error estimates are obtained for smooth solutions. Some numerical examples are given to display the performance of the LDG method.

Error Estimation of the Relaxation Finite Difference Scheme for the Nonlinear Schrödinger Equation

We consider an initial- and boundary-value problem for the nonlinear Schrödinger equation with homogeneous Dirichlet boundary conditions in the one space dimension case. We discretize the problem in space by a central finite difference method and in time by the Relaxation Scheme proposed by C. Besse [C. R. Acad. Sci. Paris Sér. I Math., 326 (1998), pp. 1427–1432]. We provide optimal order error estimates, in the discrete -norm, for the approximation error at the time nodes and at the intermediate time nodes. In the context of the nonlinear Schrödinger equation, this is the first time that the derivation of an error estimate, for a fully discrete method based on the Relaxation Scheme, is completely addressed.

Improved a priori error estimates for a discontinuous Galerkin pressure correction scheme for the Navier–Stokes equations

AbstractThe pressure correction scheme is combined with interior penalty discontinuous Galerkin method to solve the time‐dependent Navier–Stokes equations. Optimal error estimates are derived for the velocity in the L2 norm in time and in space. Error bounds for the discrete time derivative of the velocity and for the pressure are also established. The analysis is challenging and technical and is based on appropriate use of lift operators and duality arguments.

Error Analysis of Fully Discrete Scheme for the Cahn–Hilliard–Magneto-Hydrodynamics Problem

In this paper we analyze a fully discrete scheme for a general Cahn–Hilliard equation coupled with a nonsteady Magneto-hydrodynamics flow, which describes two immiscible, incompressible and electrically conducting fluids with different mobilities, fluid viscosities and magnetic diffusivities. A typical fully discrete scheme, which is comprised of conforming finite element method and the Euler semi-implicit discretization based on a convex splitting of the energy of the equation is considered in detail. We prove that our scheme is unconditionally energy stable and obtain some optimal error estimates for the concentration field, the chemical potential, the velocity field, the magnetic field and the pressure. The results of numerical tests are presented to validate the rates of convergence.

A two-grid method with backtracking for Magnetohydrodynamic equations with low electromagnetic Reynolds number

In this paper, a two-grid method with backtracking is proposed and analyzed for Magnetohydrodynamic(MHD) equations with low electromagnetic Reynolds number. This algorithm is constructed by combining the classical two-grid scheme and the backtracking technique, namely, the coarse grid correction. For the proposed algorithm, the optimal error estimates are established both in energy norm and L2 norm.

Optimal Error Estimates of Coupled and Divergence-Free Virtual Element Methods for the Poisson–Nernst–Planck/Navier–Stokes Equations and Applications in Electrochemical Systems

Optimal Error Estimates of Coupled and Divergence-Free Virtual Element Methods for the Poisson–Nernst–Planck/Navier–Stokes Equations and Applications in Electrochemical Systems

Optimal error estimates of a time-splitting Fourier pseudo-spectral scheme for the Klein–Gordon–Dirac equation

Recently, a time-splitting Fourier pseudo-spectral (TSFP) scheme for solving numerically the Klein–Gordon–Dirac equation (KGDE) has been proposed (Yi et al., 2019). However, that paper only gives numerical experiments and lacks rigorous convergence analysis and error estimates for the scheme. In addition, the time symmetry of the scheme has not been proved. This is not satisfactory from the perspective of geometric numerical integration. In this paper, we proposed a new TSFP scheme for the KGDE with periodic boundary conditions by reformulating the Klein–Gordon part into a relativistic nonlinear Schrödinger equation. The new scheme is time symmetric, fully explicit and conserves the discrete mass exactly. We make a rigorously convergence analysis and establish error estimates by comparing semi-discretization and full-discretization using the mathematical induction. The convergence rate of the scheme is proved to be second-order in time and spectral-order in space, respectively, in a generic norm under the specific regularity conditions. The numerical experiments support our theoretical analysis. The conclusion is also applicable to high-dimensional problems under sufficient regular conditions. Our scheme can also serve as a reference for solving some other coupled equations or systems such as Klein–Gordon–Schrödinger equation.

Variable-time-step BDF2 nonconforming VEM for coupled Ginzburg-Landau equations

In this paper, we solve the coupled Ginzburg-Landau equations by using a linearized variable-time-step second order backward differentiation formula in time combining with a nonconforming virtual element method in space. Based on the techniques of the discrete orthogonal convolution kernels and the discrete complementary convolution kernels, the unconditional optimal L2-norm error estimate of the fully discrete scheme was presented by using the error splitting technique under the mild restriction on the ratio of adjacent time-steps ratios. Finally, numerical experiments illustrate the correctness of our theoretical analysis.

An Interface/Boundary-Unfitted EXtended HDG Method for Linear Elasticity Problems

An interface/boundary-unfitted eXtended hybridizable discontinuous Galerkin (X-HDG) method of arbitrary order is proposed for linear elasticity interface problems on unfitted meshes with respect to the interface and domain boundary. The method uses piecewise polynomials of degrees \(k\ (\ge 1)\) and \(k-1\) respectively for the displacement and stress approximations in the interior of elements inside the subdomains separated by the interface, and piecewise polynomials of degree k for the numerical traces of the displacement on the inter-element boundaries inside the subdomains and on the interface/boundary of the domain. Optimal error estimates in \(L^2\)-norm for the stress and displacement are derived. Finally, numerical experiments confirm the theoretical results and show that the method also applies to the case of crack-tip domain.