L2 Error Estimates Research Articles

Two mixed virtual element formulations for parabolic integro-differential equations with nonsmooth initial data

This article presents and examines two distinctive approaches to the mixed virtual element method (VEM) applied to parabolic integro-differential equations (PIDEs) with non-smooth initial data. In the first part of the paper, we introduce and analyze a mixed virtual element scheme for PIDE that eliminates the need for the resolvent operator. Through the introduction of a novel projection involving a memory term, coupled with the application of energy arguments and the repeated use of an integral operator, this study establishes optimal L2-error estimates for the two unknowns p and σ. Furthermore, optimal error estimates are derived for the standard mixed formulation with a resolvent kernel. The paper offers a comprehensive analysis of the VEM, encompassing both formulations.

A stabilized finite volume method based on the rotational pressure correction projection for the time-dependent incompressible MHD equations

This paper presents an efficient numerical scheme for solving the time-dependent incompressible Magnetohydrodynamics (MHD) equations based on the rotational pressure correction projection method within the finite volume framework. Utilizing the lowest equal-order mixed finite element pair (P1−P1−P1) to approximate the velocity, magnetic and pressure fields, our numerical scheme satisfies the discrete inf–sup condition by the pressure projection stabilization. To tackle the coupling inherent in the time-dependent incompressible MHD equations,the rotational pressure correction projection method is introduced to split the original problem into several linear subproblems. The unconditional stability of numerical schemes are provided,optimal error estimates in both L2 and H1-norms of numerical solutions are also presented. Finally, some numerical results are given to verify the established theoretical findings and show the performances of the considered numerical schemes.

Optimal [formula omitted] error estimates of mass- and energy- conserved FE schemes for a nonlinear Schrödinger–type system

In this paper, we present an implicit Crank–Nicolson finite element (FE) scheme for solving a nonlinear Schrödinger–type system, which includes Schrödinger–Helmholz system and Schrödinger–Poisson system. In our numerical scheme, we employ an implicit Crank–Nicolson method for time discretization and a conforming FE method for spatial discretization. The proposed method is proved to be well-posedness and ensures mass and energy conservation at the discrete level. Furthermore, we prove optimal L2 error estimates for the fully discrete solutions. Finally, some numerical examples are provided to verify the convergence rate and conservation properties.

Uniform error analysis of an exponential IMEX-SAV method for the incompressible flows with large Reynolds number based on grad-div stabilization

Based on the grad-div stabilization and scalar auxiliary variable (SAV) methods, a first-order Euler implicit/explicit finite element scheme is studied for the Navier–Stokes equations with large Reynolds number. In the designing of numerical scheme, the nonlinear term is explicitly treated such that one only needs to solve a constant coefficient algebraic system at each time step. Meanwhile, the proposed scheme is unconditionally stable without any condition of the time step τ and mesh size h. In finite element discretization, we use the stable Taylor-Hood element for the approximation of the velocity and pressure. By a rigorous analysis, we derive an uniform L2 error estimate O(τ+h2) of the velocity in which the constant is independent of the viscosity coefficient. Finally, numerical experiments are given to support theoretical results and the efficiency of the proposed scheme.

Streamline Diffusion Weak Galerkin Finite Element Methods for Linear Unsteady State Convection Diffusion Equations and Error Analysis

In this paper, the streamline diffusion weak Galerkin finite element method is proposed and analyzed for solving unsteady time convection diffusion problem in two dimension. The v-elliptic property and the stability of this scheme are proved in terms of some conditions. We derive an error estimate in L2(μ) and H1(μ) norm. Numerical experiments have demonstrated the effectiveness of the method in solving convection propagation problems, and the theoretical analysis has been validated.

Optimal error estimates of a non-uniform IMEX-L1 finite element method for time fractional PDEs and PIDEs

Stability and optimal convergence analysis of a non-uniform implicit-explicit L1 finite element method (IMEX-L1-FEM) is studied for a class of time-fractional linear partial differential/integro-differential equations with non-self-adjoint elliptic part having (space-time) variable coefficients. The proposed scheme is based on a combination of an IMEX-L1 method on graded mesh in the temporal direction and a finite element method in the spatial direction. With the help of a discrete fractional Grönwall inequality, global almost optimal error estimates in L2- and H1-norms are derived for the problem with initial data u0∈H01(Ω)∩H2(Ω). The novelty of our approach is based on managing the interaction of the L1 approximation of the fractional derivative and the time discrete elliptic operator to derive the optimal estimate in H1-norm directly. Furthermore, a super convergence result is established when the elliptic operator is self-adjoint with time and space varying coefficients, and as a consequence, an L∞ error estimate is obtained for 2D problems that too with the initial condition is in H01(Ω)∩H2(Ω). All results proved in this paper are valid uniformly as α→1−, where α is the order of the Caputo fractional derivative. Numerical experiments are presented to validate our theoretical findings.

Optimal error estimates of a second-order fully decoupled finite element method for the nonstationary generalized Boussinesq model

In this paper, we develop and analyze a fully decoupled finite element method for the non-stationary generalized Boussinesq equations, where the viscosity and thermal conductivity depend on the temperature. Based on some subtle implicit-explicit treatments for the nonlinear coupling terms, we develop a second-order in time, fully decoupled, linear and unconditionally energy stable scheme for solving this system. The unconditional stability of the fully discrete scheme with finite element approximation is proved. The optimal L2-error estimates are analyzed for temperature-dependent thermal conductivity system. Numerical experiments are presented to illustrate the convergence, accuracy and applicability of the proposed numerical scheme.

Convergence analysis of a second order numerical scheme for the Flory–Huggins–Cahn–Hilliard–Navier–Stokes system

We present an optimal rate convergence analysis for a second order accurate in time, fully discrete finite difference scheme for the Cahn–Hilliard–Navier–Stokes (CHNS) system, combined with logarithmic Flory–Huggins energy potential. The numerical scheme has been recently proposed, and the positivity-preserving property of the logarithmic arguments, as well as the total energy stability, have been theoretically justified. In this paper, we rigorously prove second order convergence of the proposed numerical scheme, in both time and space. Since the CHNS is a coupled system, the standard ℓ∞(0,T;ℓ2)∩ℓ2(0,T;Hh2) error estimate could not be easily derived, due to the lack of regularity to control the numerical error associated with the coupled terms. Instead, the ℓ∞(0,T;Hh1)∩ℓ2(0,T;Hh3) error analysis for the phase variable and the ℓ∞(0,T;ℓ2) analysis for the velocity vector, which shares the same regularity as the energy estimate, is more suitable to pass through the nonlinear analysis for the error terms associated with the coupled physical process. Furthermore, the highly nonlinear and singular nature of the logarithmic error terms makes the convergence analysis even more challenging, since a uniform distance between the numerical solution and the singular limit values of is needed for the associated error estimate. Many highly non-standard estimates, such as a higher order asymptotic expansion of the numerical solution (up to the third order accuracy in time and fourth order in space), combined with a rough error estimate (to establish the maximum norm bound for the phase variable), as well as a refined error estimate, have to be carried out to conclude the desired convergence result. To our knowledge, it will be the first work to establish an optimal rate convergence estimate for the Cahn–Hilliard–Navier–Stokes system with a singular energy potential.

Pressure-robust [formula omitted] error analysis for Raviart–Thomas enriched Scott–Vogelius pairs

Recent work shows that it is possible to enrich the Scott–Vogelius finite element pair by certain Raviart–Thomas functions to obtain an inf–sup stable and divergence-free method on general shape-regular meshes. A skew-symmetric consistency term was suggested for avoiding an additional stabilization term for higher order elements, but no L2(Ω) error estimate was shown for the Stokes equations. This note closes this gap. In addition, the optimal choice of the stabilization parameter is studied numerically.

Adaptive guaranteed lower eigenvalue bounds with optimal convergence rates

AbstractGuaranteed lower Dirichlet eigenvalue bounds (GLB) can be computed for the m-th Laplace operator with a recently introduced extra-stabilized nonconforming Crouzeix–Raviart ($$m=1$$ m = 1 ) or Morley ($$m=2$$ m = 2 ) finite element eigensolver. Striking numerical evidence for the superiority of a new adaptive eigensolver motivates the convergence analysis in this paper with a proof of optimal convergence rates of the GLB towards a simple eigenvalue. The proof is based on (a generalization of) known abstract arguments entitled as the axioms of adaptivity. Beyond the known a priori convergence rates, a medius analysis is enfolded in this paper for the proof of best-approximation results. This and subordinated $$L^2$$ L 2 error estimates for locally refined triangulations appear of independent interest. The analysis of optimal convergence rates of an adaptive mesh-refining algorithm is performed in 3D and highlights a new version of discrete reliability.

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.

Uniform Convergence Analysis of the Discontinuous Galerkin Method on Layer-Adapted Meshes for Singularly Perturbed Problem

This paper concerns a discontinuous Galerkin (DG) method for a one-dimensional singularly perturbed problem which possesses essential characteristic of second order convection-diffusion problem after some simple transformations. We derive an optimal convergence of the DG method for eight layer-adapted meshes in a general framework. The convergence rate is valid independent of the small parameter. Furthermore, we establish a sharperL2-error estimate if the true solution has a special regular component. Numerical experiments are also given.

An operator splitting Legendre-tau spectral method for Maxwell’s equations with nonlinear conductivity in two dimensions

In the paper, an operator splitting Legendre-tau spectral method is proposed to solve Maxwell’s equations with nonlinear conductivity in two dimensions. By the implicit–explicit numerical scheme, the linear and nonlinear parts are treated in different ways. The linear terms are treated by the Legendre-tau spectral method implicitly, and the nonlinear terms are treated by some collocation methods explicitly. The scheme uses polynomial spaces of different degrees to approximate the electric and magnetic fields, respectively, so that they can be decoupled in computation. The splitting leap frog Crank–Nicolson method is applied in time discretization, which is a three-level scheme with the nonlinear term being treated explicitly in intermediate level. With the application of inverse inequality, for the Legendre interpolation operator, optimal L2-error estimates are achieved for the proposed scheme. In numerical examples, the nonlinear terms are computed at Chebyshev–Gauss–Lobatto points by the fast Legendre transform. Numerical results validate the efficiency and spectral accuracy of the scheme.

L2 error estimates for a family of cubic finite volume methods on triangular meshes

This paper considers the error estimates of a family of Lagrange cubic finite volume methods for solving two dimensional elliptic boundary value problems over triangular meshes. By introducing a novel map of finite volume method, we propose a family of cubic finite volume method schemes with four independent map coefficients. It is detailedly proved that the proposed finite volume method schemes is exactly equivalent to the existing traditional finite volume method schemes of the same boundary value problems. A geometric constraint requirement on the triangular meshes of primary partitions is imposed to guarantee the uniform ellipticity of discrete elliptic bilinear forms resulted from the proposed schemes. The geometric requirements of the primary partitions could be greatly attenuated when the map coefficients are appropriately designed. Once the uniform ellipticity of the discrete elliptic bilinear forms is established, the proposed schemes have the unique solvability, and the optimal H1-norm error estimates without any restriction on dual elements of dual partitions of the finite volume methods, which is consistent with the results of literatures. With the help of the introduced map, the proposed schemes have the optimal L2-norm error estimates with a mesh restriction for the dual elements of their dual partitions significantly weaker than the existing theoretical results, in which the Aubin-Nitsche duality argument and the orthogonal conditions based on the dual partitions are employed. Finally, we validate our theoretical results by numerical experiments.

Analysis of the MAC scheme for the three dimensional Stokes problem

In this paper, we propose a marker and cell method (MAC) with a proper quadrature scheme over non-uniform cuboid grids by using a staggered finite volume element method (FVEM) for 3D Stokes equations. The stability of the proposed MAC scheme is proven under Petrov-Galerkin method. By establishing a connection between MAC and FVEM, we rigorously prove the superconvergence property and the optimal order L2 error estimate. Finally, numerical results verify our theoretical findings.

One-parameter spectral Galerkin methods for Timoshenko beam system with delay boundary feedback

This paper deals with Timoshenko beam (TB) system with delay boundary feedback (DBF). For external-force-free TB system with DBF, an energy stability criterion is established. For solving general TB system with DBF, a class of one-parameter spectral Galerkin (OPSG) methods with parameter θ∈[0,1] are suggested. It is proved under the appropriate conditions that OPSG methods can preserve the energy stability in the discrete sense. Based on a new projection operator and some analytical techniques, an L2-error estimate of the methods is derived. Finally, by performing several numerical experiments, the obtained theoretical results and the computational accuracy of the methods are further illustrated.

A new error analysis of backward Euler Galerkin finite element method for two-dimensional time-dependent Ginzburg–Landau equation

In this paper, the backward Euler Galerkin finite element method (FEM) is investigated for the two dimensional time-dependent Ginzburg–Landau equation. The unconditionally optimal L2 error estimate is obtained without using the boundedness of the numerical solution in L∞ norm, while it is an indispensable requirement in the previous works. A key to the analysis is to deal with the nonlinear term rigorously and skillfully in terms of the boundedness of the numerical solution in L2 norm rather than the L∞ norm. Numerical results are presented to confirm the theoretical findings.

Convergence rates in homogenization of parabolic systems with locally periodic coefficients

In this paper we study the quantitative homogenization of second-order parabolic systems with locally periodic (in both space and time) coefficients. The O(ε) scale-invariant error estimate in L2(0,T;L2dd−1(Ω)) is established in C1,1 cylinders under minimum smoothness conditions on the coefficients. This process relies on critical estimates of smoothing operators. We also develop a new construction of flux correctors in the parabolic manner and a sharp estimate for temporal boundary layers.

Generalized fractional-order Legendre wavelet method for two dimensional distributed order fractional optimal control problem

This paper is concerned with a two-dimensional fractional optimal control problem whose governing equations are distributed order fractional differential equations in the Caputo sense. A generalized fractional-order Legendre wavelet method has been used to solve the two-dimensional distributed-order fractional optimal control problem. An exact formula for the Riemann–Liouville integration of generalized fractional-order Legendre wavelet has been derived by using regularized beta functions. This formula and the two-dimensional Gauss–Legendre integration formula have been used to solve the two-dimensional distributed order fractional optimal control problem. Moreover, an L2-error estimate in the approximation of an unknown function with a generalized fractional-order Legendre wavelet has been derived and the estimated order has been verified for a given function. Furthermore, convergence analysis for the proposed method has been presented. In the last, two test problems have been considered to illustrate the efficiency of the proposed method.

A New Approach to Handle Curved Meshes in the Hybrid High-Order Method

We present here a novel approach to handling curved meshes in polytopal methods within the framework of hybrid high-order methods. The hybrid high-order method is a modern numerical scheme for the approximation of elliptic PDEs. An extension to curved meshes allows for the strong enforcement of boundary conditions on curved domains and for the capture of curved geometries that appear internally in the domain e.g. discontinuities in a diffusion coefficient. The method makes use of non-polynomial functions on the curved faces and does not require any mappings between reference elements/faces. Such an approach does not require the faces to be polynomial and has a strict upper bound on the number of degrees of freedom on a curved face for a given polynomial degree. Moreover, this approach of enriching the space of unknowns on the curved faces with non-polynomial functions should extend naturally to other polytopal methods. We show the method to be stable and consistent on curved meshes and derive optimal error estimates in L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document} and energy norms. We present numerical examples of the method on a domain with curved boundary and for a diffusion problem such that the diffusion tensor is discontinuous along a curved arc.