H1 Error Estimates Research Articles

A novel family of Q1-finite volume element schemes on quadrilateral meshes

A novel family of isoparametric bilinear finite volume element schemes are constructed and analyzed to solve the anisotropic diffusion problems on general convex quadrilateral meshes. These new schemes are obtained by employing a special quadrature rule to approximate the line integrals in classical Q1-finite volume element method. The new quadrature rule is a linear combination of trapezoidal and midpoint rules, and the weights depend on a parameter ωK. The novelty of this work is that, for any fully anisotropic diffusion tensor, we provide some specific ωK to ensure the coercivity result of the proposed schemes on arbitrary parallelogram, quasi-parallelogram, trapezoidal and some general convex quadrilateral meshes. More interesting is that, the parameter ωK can only involves the anisotropic diffusion tensor and the geometry of quadrilateral cell. An optimal H1 error estimate is also proved on quasi-parallelogram meshes. Finally, the theoretical findings are validated by several numerical examples.

Two-grid methods for nonlinear pseudo-parabolic integro-differential equations by finite element method

In this paper, two efficient two-grid finite element algorithms are proposed for solving two-dimensional nonlinear pseudo-parabolic integro-differential equations. Firstly, we obtain optimal error estimates of the fully discrete finite element method using a temporal-spatial error splitting technique in H1 and Lp norms. Then the two-grid technique to improve computation efficiency of the proposed finite element method. Error estimates in H1 and Lp norms of two-grid solutions are presented. Theoretical analysis shows that the two-grid algorithms maintain asymptotically optimal accuracy. Finally, numerical examples are provided to support our theoretical results and demonstrate the effectiveness of these methods.

A parallel finite element post-processing algorithm for the damped Stokes equations

This paper presents and analyzes a parallel finite element post-processing algorithm for the simulation of Stokes equations with a nonlinear damping term, which integrates the algorithmic advantages of the two-level approach, the partition of unity method and post-processing technique. The most valuable highlights of the present algorithm are that (1) a global continuous approximate solution is generated via the partition of unity method; (2) by adding an extra coarse grid correction step, the smoothness of the approximate solution is improved; (3) it has a good parallel performance since there requires little communication in solving a series of residual problems in the subdomain of interest. We theoretically derive the L2-error estimates both for the approximate velocity and pressure and H1-error estimate for the velocity under some necessary conditions. Meanwhile, we numerically perform various test examples to validate the theoretically predicted convergence rate and illustrate the high efficiency of the proposed algorithm.

Long time [formula omitted]-stability of fast L2-1[formula omitted] method on general nonuniform meshes for subdiffusion equations

In this work, the global-in-time H1-stability of a fast L2-1σ method on general nonuniform meshes is studied for subdiffusion equations, where the convolution kernel in the Caputo fractional derivative is approximated by sum of exponentials. Under some mild restrictions on time stepsize, a bilinear form associated with the fast L2-1σ formula is proved to be positive semidefinite for all time. As a consequence, the uniform global-in-time H1-stability of the fast L2-1σ schemes can be derived for both linear and semilinear subdiffusion equations, in the sense that the H1-norm is uniformly bounded as the time tends to infinity. To the best of our knowledge, this appears to be the first work for the global-in-time H1-stability of fast L2-1σ scheme on general nonuniform meshes for subdiffusion equations. Moreover, the sharp finite time H1-error estimate for the fast L2-1σ schemes is reproved based on more delicate analysis of coefficients where the restriction on time step ratios is relaxed comparing to existing works.

A variable time-step IMEX-BDF2 SAV scheme and its sharp error estimate for the Navier–Stokes equations

We generalize the implicit-explicit (IMEX) second-order backward difference (BDF2) scalar auxiliary variable (SAV) scheme for Navier–Stokes equation with periodic boundary conditions (Huang and Shen, SIAM J. Numer. Anal. 59 (2021) 2926–2954) to a variable time-step IMEX-BDF2 SAV scheme, and carry out a rigorous stability and convergence analysis. The key ingredients of our analysis are a new modified discrete Grönwall inequality, exploration of the discrete orthogonal convolution (DOC) kernels, and the unconditional stability of the proposed scheme. We derive global and local optimal H1 error estimates in 2D and 3D, respectively. Our analysis provides a theoretical support for solving Navier–Stokes equations using variable time-step IMEX-BDF2 SAV schemes. We also design an adaptive time-stepping strategy, and provide ample numerical examples to confirm the effectiveness and efficiency of our proposed methods.

Error estimate of the cell-centered nonlinear positivity-preserving two-point flux approximation schemes

We present the error estimate of nonlinear positivity-preserving finite volume schemes for diffusion problems. These schemes are also named by nonlinear two-point flux approximation (NTPFA) methods. Due to their conservation and monotonicity, such schemes have attracted wide attention. Under suitable regularity assumptions on the exact solution, we prove a discrete H1 error estimate through the analysis of consistent errors. Some numerical examples are presented to demonstrate the theoretical results.

Two-grid virtual element discretization of semilinear elliptic problem

In this paper a two grid algorithm for semilinear elliptic problem based on virtual element method (VEM) discretization is proposed. With this new algorithm the solution of a semilinear elliptic problem on a fine grid is reduced to the solution of a semilinear elliptic problem on a much coarser grid, and the solution of a linear system on the fine grid. A priori error estimates in H1 and L2 norms are derived. Numerical experiments are carried out to illustrate the theoretical findings.

Unconditionally optimal H1-error estimate of a fast nonuniform L2-1σ scheme for nonlinear subdiffusion equations

Unconditionally optimal H1-error estimate of a fast nonuniform L2-1σ scheme for nonlinear subdiffusion equations

A two-grid method for discontinuous Galerkin approximations to compressible miscible displacement problems

Discontinuous Galerkin approximations are employed to the compressible miscible displacement problem. A two-grid algorithm is proposed, which is based on one coarse grid space and one fine grid space. The H1 error estimate of the concentration and the L2 error estimate of the velocity are presented for the proposed two-grid method, which show that the two-grid method achieves optimal approximations if the mesh sizes satisfy h=O(H2). The numerical results are presented to confirm that the algorithm is effective.

A conforming discontinuous Galerkin finite element method for elliptic interface problems

A new conforming discontinuous Galerkin method, which is based on weak Galerkin finite element method, is introduced for solving second order elliptic interface problems with discontinuous coefficient. The numerical method studied in this paper has no stabilizer and fewer unknowns compared with the known weak Galerkin algorithms. The error estimates in H1 and L2 norms are established, which are the optimal order convergence. Numerical experiments demonstrate the performance of the method, confirm the theoretical results of accuracy.

Unfitted hybrid high-order methods for the wave equation

We design an unfitted hybrid high-order (HHO) method for the wave equation. The wave propagates in a domain where a curved interface separates subdomains with different material properties. The key feature of the discretization method is that the interface can cut more or less arbitrarily through the mesh cells. We address both the second-order formulation in time of the wave equation and its reformulation as a first-order system. We prove H1-error estimates for a space semi-discretization in space of the second-order formulation, leading to optimal convergence rates for smooth solutions. Numerical experiments illustrate the theoretical findings and show that the proposed numerical schemes can be used to simulate accurately the propagation of acoustic waves in heterogeneous media, with meshes that are not fitted to the geometry. For the second-order formulation, the implicit, second-order accurate Newmark scheme is used for the time discretization, whereas (diagonally-implicit or explicit) Runge–Kutta schemes up to fourth-order accuracy are used for the first-order formulation. In the explicit case, we study the CFL condition on the time step and observe that the unfitted approach combined with local cell agglomeration leads to a comparable condition as when using fitted meshes.

A family of quadratic finite volume element schemes for anisotropic diffusion problems on triangular meshes

In this paper, we propose and analyze a family of quadratic finite volume element schemes for anisotropic diffusion problems on triangular meshes. The present work covers a previous one Zhou and Wu (2020) where only the case of scalar diffusion coefficients is considered. The novelty of this paper is that, by the introduction of some special parameters involving the diffusion tensor and the mesh geometry, each entry of the element stiffness matrix can be split as three parts that are easily computed and analyzed. Thanks to this representation, we manage to obtain a sufficient condition that ensures the existence, uniqueness and coercivity result of the finite volume element solution on triangular mesh. Moreover, this sufficient condition has a simple, analytic and computable expression that can be easily judged on any triangular meshes with arbitrary full diffusion tensors. In particular, when the diffusion tensor is an identity matrix, this condition reduces to a geometric one and only relies on the interior angles of each triangular element, which is consistent with the result in (Zhou and Wu, 2020). Moreover, based on this sufficient condition, we obtain a minimum angle condition for the case of general diffusion tensors, which is better than some existing ones. By the coercivity result, the optimal H1 error estimate is also obtained. Finally, the theoretical results are verified by some numerical experiments.

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 unified analysis of a class of quadratic finite volume element schemes on triangular meshes

This paper presents a general framework for the coercivity analysis of a class of quadratic finite volume element (FVE) schemes on triangular meshes for solving elliptic boundary value problems. This class of schemes covers all the existing quadratic schemes of Lagrange type. With the help of a new mapping from the trial function space to the test function space, we find that each element matrix can be decomposed into three parts: the first part is the element stiffness matrix of the standard quadratic finite element method (FEM), the second part is the difference between the FVE and FEM on the element boundary, while the third part can be expressed as the tensor product of two vectors. Thanks to this decomposition, we obtain a sufficient condition to guarantee the existence, uniqueness, and coercivity result of the FVE solution on triangular meshes. Moreover, based on this sufficient condition, some minimum angle conditions with simple, analytic, and computable expressions can be derived and they depend only on the constructive parameters of the schemes. As a byproduct, some existing coercivity results are improved. Finally, an optimal H1 error estimate is proved by the standard techniques.

The H1-error analysis of the finite element method for solving the fractional diffusion equation

In this paper, we study the fully discrete numerical method for solving the time-fractional diffusion equation: ∂tαu−div(A∇u)=f, 0<α<1. This numerical method is based on the L1 difference approximation to ∂tαu and the r-order finite element discretization on the spatial domain. We first establish a new error bound of O(△tβ)-order for the L1 formula where β (1−α≤β≤2−α) depends on the smoothness of function u(t). Then, we show that this fully discrete scheme is H1-unconditionally stable and the discrete solution admits the optimal H1-error estimate of O(△tβ+hr)-order. When the exact solution u(t) is weakly singular at t=0, by using the graded time-meshes, we also derive the H1-stability and the optimal H1-error estimate. Moreover, by using the postprocessing technique, we derive an H1-superconvergence estimate of O(△tβ+hr+1)-order. Numerical examples are provided to support our theoretical analysis.

An extended P1-nonconforming finite element method on general polytopal partitions

An extended P1-nonconforming finite element method is developed in this article for the Dirichlet boundary value problem of convection–diffusion–reaction equations on general polytopal partitions. This new method was motivated by the simplified weak Galerkin method, and makes use of only the degrees of freedom on the boundary of each element and, hence, has reduced computational complexity. Numerical stability and optimal order of error estimates in H1 and L2 norms are established for the corresponding numerical solutions. Some numerical results are presented to computationally verify the mathematical convergence theory. A superconvergence phenomenon on rectangular partitions is noted and illustrated through various numerical experiments.

Two energy-conserving and compact finite difference schemes for two-dimensional Schrödinger-Boussinesq equations

In this paper, we study two compact finite difference schemes for the Schrodinger-Boussinesq (SBq) equations in two dimensions. The proposed schemes are proved to preserve the total mass and energy in the discrete sense. In our numerical analysis, besides the standard energy method, a “cut-off” function technique and a “lifting” technique are introduced to establish the optimal H1 error estimates without any restriction on the grid ratios. The convergence rate is proved to be of O(τ2 + h4) with the time step τ and mesh size h. In addition, a fast finite difference solver is designed to speed up the numerical computation of the proposed schemes. The numerical results are reported to verify the error estimates and conservation laws.

Analysis of the decoupled and positivity-preserving DDFV schemes for diffusion problems on polygonal meshes

We propose and analyze a family of decoupled and positivity-preserving discrete duality finite volume (DDFV) schemes for diffusion problems on general polygonal meshes. These schemes are further extensions of the scheme in Su et al. (J. Comput. Phys.372, 773–798, 2018) and share some benefits. First, the two sets of finite volume (FV) equations are constructed for the vertex-centered and cell-centered unknowns, respectively, and these two sets of equations can be solved in a decoupled way. Second, unlike most existing nonlinear positivity-preserving schemes, nonlinear iteration methods are not required for linear problems and the nonlinear solver can be selected unrestrictedly for nonlinear problems. Moreover, the positivity and well-posedness of the solution can be analyzed rigorously for linear problems. Under some weak geometric assumptions, the stability and error estimate results for the vertex-centered unknowns are obtained. Since the present DDFV schemes and the pure vertex-centered scheme in Wu et al. (Int. J. Numer. Meth. Fluids81(3), 131–150, 2016) share the same FV equations for the vertex-centered unknowns, this part itself can also serve as an analysis for the scheme in Wu et al. (Int. J. Numer. Meth. Fluids81(3), 131–150, 2016). Furthermore, by assuming the coercivity of the FV equations for the cell-centered unknowns, a first-order H1 error estimate is obtained for the cell-centered unknowns through the analysis of residual errors. Numerical experiments demonstrate both the accuracy and the positivity of discrete solutions on general grids. The efficiency of the Newton method is emphasized by comparison with the fixed-point method, which illustrates the advantage of our schemes when dealing with nonlinear problems.

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 linearly-implicit and conservative Fourier pseudo-spectral method for the 3D Gross–Pitaevskii equation with angular momentum rotation

In this paper, a linearly-implicit Fourier pseudo-spectral method which preserves discrete mass and energy is developed for the time-dependent 3D Gross–Pitaevskii equation with additional angular momentum rotation. By establishing several discrete semi-norm equivalences between the Fourier pseudo-spectral method and the finite difference method, we establish an optimal H1-error estimate for the proposed scheme without any restrictions on the grid ratio. The convergent rate of the numerical solution is proved to be of order O(N−r+τ2), where N is the number of spatial nodes and τ is the time step. Numerical results are reported to verify the efficiency and accuracy of our new method.