Error Estimates In Norm Research Articles

A Predictor-Corrector Finite Element Method for Time-Harmonic Maxwell’s Equations in Polygonal Domains

The overall efficiency and accuracy of standard finite element methods may be severely reduced if the solution of the boundary value problem entails singularities. In the particular case of time-harmonic Maxwell’s equations in nonconvex polygonal domains Ω, H1-conforming nodal finite element methods may even fail to converge to the physical solution. In this paper, we present a new nodal finite element adaptation for solving time-harmonic Maxwell’s equations with perfectly conducting electric boundary condition in general polygonal domains. The originality of the present algorithm lies in the use of explicit extraction formulas for the coefficients of the singularities to define an iterative procedure for the improvement of the finite element solutions. A priori error estimates in the energy norm and in the L2 norm show that the new algorithm exhibits the same convergence properties as it is known for problems with regular solutions in the Sobolev space H2Ω2 in convex and nonconvex domains without the use of graded mesh refinements or any other modification of the bilinear form or the finite element spaces. Numerical experiments that validate the theoretical results are presented.

Stabilizer-free weak Galerkin methods for monotone quasilinear elliptic PDEs

In this paper, we study the stabilizer-free weak Galerkin methods on polytopal meshes for a class of second order elliptic boundary value problems of divergence form and with gradient nonlinearity in the principal coefficient. With certain assumptions on the nonlinear coefficient, we show that the discrete problem has a unique solution. This is achieved by showing that the associated operator satisfies certain continuity and monotonicity properties. With the help of these properties, we derive optimal error estimates in the energy norm. We present several numerical examples to verify the error estimates.

Discontinuous Galerkin methods for fractional elliptic problems

The aim of this paper is to provide a mathematical framework for studying different versions of discontinuous Galerkin (DG) approaches for solving 2D Riemann–Liouville fractional elliptic problems on a finite domain. The boundedness and stability analysis of the primal bilinear form are provided. A priori error estimate under energy norm and optimal error estimate under $$L^{2}$$ norm are obtained for DG methods of the different formulations. Finally, the performed numerical examples confirm the optimal convergence order of the different formulations.

A Monolithic Arbitrary Lagrangian–Eulerian Finite Element Analysis for a Stokes/Parabolic Moving Interface Problem

In this paper, an arbitrary Lagrangian–Eulerian (ALE)—finite element method (FEM) is developed within the monolithic approach for a moving-interface model problem of a transient Stokes/parabolic coupling with jump coefficients—a linearized fluid-structure interaction (FSI) problem. A new $$H^1$$-projection is defined for this problem for the first time to account for the mesh motion due to the moving interface. The well-posedness and optimal convergence properties in both the energy norm and $$L^2$$ norm are analyzed for this mixed-type $$H^1$$-projection, with which the stability and optimal error estimate in the energy norm are derived for both semi- and fully discrete mixed finite element approximations to the Stokes/parabolic interface problem. Numerical experiments are carried out to validate all theoretical results. The developed analytical approach can be extended to a general FSI problem.

High order symmetric direct discontinuous Galerkin method for elliptic interface problems with fitted mesh

In this article, we aim to develop a high order symmetric direct discontinuous Galerkin method solving elliptic interface problems with zero or non-zero solution jump and flux jump interface conditions. We focus on the case, in which the mesh is partitioned along the curved interface. The two interface jump conditions are naturally and simultaneously built into the numerical flux definition on the curved triangular elements' edges that overlap with the interface. We obtain a stable and high order method, regardless of the combination of the two interface jump conditions. The two interface jump conditions are both weakly enforced in the scheme formulation. Optimal (k+1)th order L2 norm error estimate is proved for polygonal interfaces. A sequence of numerical examples is carried out to verify the optimal convergence of the symmetric direct discontinuous Galerkin method with high order P2, P3 and P4 approximations. Uniform optimal convergence order, which is independent of the diffusion coefficient ratio inside and outside of the interface, is obtained. The symmetric direct discontinuous Galerkin method is shown to be capable of handling interface problems with complicated geometries.

Characteristic mixed volume element for compressible two-phase displacement in porous media

Numerical simulation of a two-phase compressible displacement problem is considered in this paper, modelled by a nonlinear system of PDEs. Considering the characters of mathematical model, we present an efficient combination algorithm of the method of characteristics (MOC) and mixed finite volume element (MFVE). The MFVE is used for computing the pressure, Darcy velocity and the saturation. Furthermore, the saturation equation is convection-dominated. MOC eliminates numerical dispersion and nonphysical oscillation, and computes the values at the sharp fronts well. An MFVE-MOC is formed for the saturation. An optimal error estimates in l2 norm is concluded. Finally, numerical examples are given to show the effectiveness and practicability. The composite scheme possibly solves this actual problem well.

A Sequential Least Squares Method for Poisson Equation Using a Patch Reconstructed Space

We propose a new least squares finite element method to solve the Poisson equation. By using a piecewisely irrotational space to approximate the flux, we split the classical method into two sequential steps. The first step gives the approximation of flux in the new approximation space and the second step can use flexible approaches to give the pressure. The new approximation space for flux is constructed by patch reconstruction with one unknown per element consisting of piecewisely irrotational polynomials. The error estimates in the energy norm and $L^2$ norm are derived for the flux and the pressure. Numerical results verify the convergence order in error estimates, and demonstrate the flexibility and particularly the great efficiency of our method.

Arbitrary Lagrangian--Eulerian Discontinuous Galerkin Method for Hyperbolic Equations Involving $\delta$-Singularities

In this paper, we develop and analyze an arbitrary Lagrangian--Eulerian discontinuous Galerkin (ALE-DG) method for solving one-dimensional hyperbolic equations involving $\delta$-singularities on moving meshes. The $L^2$ and negative norm error estimates are proven for the ALE-DG approximation. More precisely, when choosing the approximation space with piecewise $k$th degree polynomials, the convergence rate in $L^2$-norm for the scheme with the upwind numerical flux is $(k+1)$th order in the region apart from the singularities, the convergence rate in $H^{-(k+1)}$ norm for the scheme with the monotone fluxes in the whole domain is $k$th order, the convergence rate in $H^{-(k+2)}$ norm for the scheme with the upwind flux in the whole domain can achieve $(k+\frac{1}{2})$th order, and the convergence rate in $H^{-(k+1)}(R\backslash R_T)$ norm for the scheme with the upwind flux is $(2k+1)$th order, where $R_T$ is the pollution region at time $T$ due to the singularities. Moreover, numerically the $(2k+1)$th order accuracy for the postprocessed solution in the smooth region can be obtained, which is produced by convolving the ALE-DG solution with a suitable kernel consisting of B-splines. Numerical examples are shown to demonstrate the accuracy and capability of the ALE-DG method for the hyperbolic equations involving $\delta$-singularity on moving meshes.

A New Hybrid Staggered Discontinuous Galerkin Method on General Meshes

A novel high-order hybrid staggered discontinuous Galerkin method for general meshes is proposed to solve general second order elliptic problems. Our new formulation is related to standard staggered discontinuous Galerkin method, but more flexible and cost effective: rough grids are allowed and the size of the final system is remarkably reduced thanks to the partial hybridization. Optimal convergence estimates for both the scalar and vector variables are developed. Moreover, superconvergent results with respect to discrete $$H^1$$ norm and $$L^2$$ norm for the scalar variable are proved and negative norm error estimates for both the scalar and vector variables are also developed. On the other hand, mesh adaptation is particularly simple since hanging nodes are allowed, which makes the proposed method well suited for adaptive mesh refinement. Therefore, we design a residual type a posteriori error estimator, and the reliability and local efficiency of the error estimator are proved. Numerical experiments confirm the theoretical findings.

A second order numerical scheme for nonlinear Maxwell’s equations using conforming finite element

In this paper, we study and analyze a second order numerical scheme for Maxwell’s equations with nonlinear conductivity, using the Nédelec Finite Element Method (FEM). A purely explicit treatment of the nonlinear term greatly simplifies the computational effort, since we only need to solve a constant-coefficient linear system at each time step. The curl-conforming nature of the Nédelec element assures its divergence-free property. In turn, we present the linearized stability analysis for the numerical error function to obtain an optimal L2 error estimate. In more details, an O(τ2+hs) error estimate in the L2 norm yields the maximum norm bound of the numerical solution, so that the convergence analysis could be carried out at the next time step. A few numerical examples in the transverse electric (TE) case in two dimensional spaces are also presented, which demonstrate the efficiency and accuracy of the proposed numerical scheme.

The error analysis of linearized discontinuous Galerkin finite element method for incompressible miscible displacement in porous media

We present a linearization scheme for discontinuous Galerkin method for solving the fully coupled one-phase flow problem. The pressure and concentration equations are solved by symmetric interior penalty Discontinuous Galerkin (SIPG) method and we discretized the model using (DGM) space and backward Euler in time. We prove the error estimate in the energy norm.

On subgrid multiscale stabilized finite element method for advection-diffusion-reaction equation with variable coefficients

In this study a stabilized finite element method for solving advection-diffusion-reaction equation with spatially variable coefficients has been introduced. Here subgrid scale approach along with algebraic approximation to the sub-scales has been chosen to stabilize the Galerkin finite element method. Both a priori and a posteriori finite element error estimates in L2 norm have been derived after introducing the stabilized variational form. An expression of the stabilization parameter has also been derived here. At last numerical experiments are presented to verify numerical performance of the stabilized method and the credibility of the theoretically derived expression of the stabilization parameter has been established numerically.

Numerical analysis for the Cahn-Hilliard-Hele-Shaw system with variable mobility and logarithmic Flory-Huggins potential

In this paper, a fully discrete scheme is proposed for solving the Cahn-Hilliard-Hele-Shaw system with a logarithmic potential and concentration dependent mobility. Using the regularization procedure, the domain for the logarithmic free energy density function F(ϕ) is extended from (−1,1) to (−∞,∞). A convex-splitting method of the energy is adopted in time and the mixed finite element method is used in space. Furthermore, we prove the stability of the proposed numerical method and establish the optimal error estimate in the energy norm. Finally, numerical results are presented to support our theoretical analysis.

A stabilized cut streamline diffusion finite element method for convection–diffusion problems on surfaces

We develop a stabilized cut finite element method for the stationary convection–diffusion problem on a surface embedded in Rd. The cut finite element method is based on using an embedding of the surface into a three dimensional mesh consisting of tetrahedra and then using the restriction of the standard piecewise linear continuous elements to a piecewise linear approximation of the surface. The stabilization consists of a standard streamline diffusion stabilization term on the discrete surface and a so called normal gradient stabilization term on the full tetrahedral elements in the active mesh. We prove optimal order a priori error estimates in the standard norm associated with the streamline diffusion method and bounds for the condition number of the resulting stiffness matrix. The condition number is of optimal order for a specific choice of method parameters. Numerical examples supporting our theoretical results are also included.

An interface penalty finite element method for elliptic interface problems on piecewise meshes

An interface penalty finite element method (IPFEM) is proposed for elliptic interface problems, which allows to use different meshes in different subdomains separated by the interface. The transmission conditions across the interface are treated by the Nitsche’s method (or penalty technique) with some harmonic-weighted averages. Both symmetric IPFEM and non-symmetric IPFEM are analyzed. Optimal order error estimates in energy norm and L2-norm and the flux error estimate in L2-norm are proved for the symmetric IPFEM. In particular, the relative error estimate in energy norm and the flux error estimate in L2-norm are independent of the ratio of mesh sizes and the contrast of the discontinuous coefficients across the interface. Error estimates for the non-symmetric IPFEM are also obtained. Furthermore, optimal mesh sizes with respect to the error estimate in energy norm and the flux error estimate in L2-norm are discussed, respectively. Numerical examples are also provided to confirm the theoretical results and show that the IPFEM on a mesh with optimal mesh sizes may give much better approximations than the one on the quasi-uniform mesh with the same number of nodal points.

Analysis of the shifted boundary method for the Stokes problem

The analysis of stability and accuracy of the shifted boundary method is developed for the Stokes flow equations. The key feature of the shifted boundary method, an embedded finite element method, is the shifting of the location where boundary conditions are applied from the true to a surrogate boundary, and an appropriate modification (shifting) of the value of the boundary conditions. An inf–sup condition is proved for the variational formulation associated to the shifted boundary method and we derive, by way of Strang’s second lemma, an optimal error estimate in the natural SBM norm. We also derive an L2-error estimate for the velocity field, by means of an extension of the Aubin–Nitsche approach to embedded, non-consistent, mixed finite element methods.

Error estimates of a conservative finite difference Fourier pseudospectral method for the Klein–Gordon–Schrödinger equation

A semi-linearized time-stepping method based on the finite difference in time and Fourier pseudospectral discretization for spatial derivatives is constructed and analyzed for the Klein–Gordon–Schrödinger equation. The resulting numerical scheme is proved to conserve the total mass and energy in the discrete levels. The maximum norm error estimates reflecting the second-order accuracy in time and spectral accuracy in space are established by using the standard energy method coupled with the induction argument. An efficient numerical algorithm is reported to speed up the evaluation of resulting algebra equation by means of the fast discrete Fourier transform. Numerical experiments are presented to show the effectiveness of our method and to confirm our analysis.

Adaptive finite element analysis with local mesh refinement based on a posteriori error estimate of element energy projection technique

PurposeThis study aims to propose a general and efficient adaptive strategy with local mesh refinement for two-dimensional (2D) finite element (FE) analysis based on the element energy projection (EEP) technique.Design/methodology/approachIn view of the inflexibility of the existing global dimension-by-dimension (D-by-D) recovery method via EEP technique, in which displacements are recovered through element strips, an improved element D-by-D recovery strategy was proposed, which enables the EEP recovery of super-convergent displacements to be implemented mostly on a single element. Accordingly, a posteriori error estimate in maximum norm was established and an EEP-based adaptive FE strategy of h-version with local mesh refinement was developed.FindingsRepresentative numerical examples, including stress concentration and singularity problems, were analyzed; the results of which show that the adaptively generated meshes reasonably reflect the local difficulties inherent in the physical problems and the proposed adaptive analysis can produce FE displacement solutions satisfying the user-specified tolerances in maximum norm with an almost optimal adaptive convergence rate.Originality/valueThe proposed element D-by-D recovery method is a more efficient and flexible displacement recovery method, which is implemented mostly on a single element. The EEP-based adaptive FE analysis can produce displacement solutions satisfying the specified tolerances in maximum norm with an almost optimal convergence rate and thus can be expected to apply to other 2D problems.

Least-Squares Methods for Elasticity and Stokes Equations with Weakly Imposed Symmetry

Abstract This paper develops and analyzes two least-squares methods for the numerical solution of linear elasticity and Stokes equations in both two and three dimensions. Both approaches use the L 2 {L^{2}} norm to define least-squares functionals. One is based on the stress-displacement/velocity-rotation/vorticity-pressure (SDRP/SVVP) formulation, and the other is based on the stress-displacement/velocity-rotation/vorticity (SDR/SVV) formulation. The introduction of the rotation/vorticity variable enables us to weakly enforce the symmetry of the stress. It is shown that the homogeneous least-squares functionals are elliptic and continuous in the norm of H ⁢ ( div ; Ω ) {H(\mathrm{div};\Omega)} for the stress, of H 1 ⁢ ( Ω ) {H^{1}(\Omega)} for the displacement/velocity, and of L 2 ⁢ ( Ω ) {L^{2}(\Omega)} for the rotation/vorticity and the pressure. This immediately implies optimal error estimates in the energy norm for conforming finite element approximations. As well, it admits optimal multigrid solution methods if Raviart–Thomas finite element spaces are used to approximate the stress tensor. Through a refined duality argument, an optimal L 2 {L^{2}} norm error estimates for the displacement/velocity are also established. Finally, numerical results for a Cook’s membrane problem of planar elasticity are included in order to illustrate the robustness of our method in the incompressible limit.

A Leap-Frog Finite Difference Method for Strongly Coupled System from Sweat Transport in Porous Textile Media

In this paper, we present an uncoupled leap-frog finite difference method for the system of equations arising from sweat transport through porous textile media. Based on physical mechanisms, the sweat transport can be viewed as the multicomponent flow that coupled the heat and moisture transfer, such that the system is nonlinear and strongly coupled. The leap-frog method is proposed to solve this system, with the second order accuracy in both spatial and temporal directions. We prove the existence and uniqueness of the solution to the system with optimal error estimates in the discrete L2 norm. Numerical simulations are presented and analyzed, respectively.