Energy Norm Research Articles

Error estimations in the balanced norm of finite element method on Bakhvalov–Shishkin triangular mesh for reaction–diffusion problems

A balanced norm, rather than the common energy norm, is introduced to reflect the behavior of layers more accurately in the finite element method for singularly perturbed reaction–diffusion problems. Convergence of optimal order in the balanced norm has been proved in the case of rectangular finite elements. However, for triangular finite elements Pk (k≥2), it is still open to prove this convergence result. With the help of the L2-stability of a weighted L2 projection, instead of the L∞-stability widely used in existing references, the geometric constraints on meshes are relaxed. As a result, the optimal order convergence in the balanced norm is proved in the case of Bakhvalov–Shishkin triangular meshes. Numerical experiments support theoretical results.

Finite element method on Shishkin mesh for a singularly perturbed problem with an interior layer

In this paper, we discuss a singularly perturbed convection diffusion problem, whose solution contains an interior layer caused by the discontinuity of the convection coefficient. For this kind of problem, uniform convergence of the finite element method has not been established. For this aim, after proving the minimum principle and stability result, we derive the asymptotic expansion of the solution and a priori estimates of its smooth and layer components. Then we construct a finite element method on Shishkin mesh, and prove uniform convergence of almost order k in the energy norm, where k is the order of piecewise polynomials.

Finite element modelling of traction-free cracks: Benchmarking the augmented finite element method (AFEM)

This paper investigates the accuracy and the convergence properties of the augmented finite element method (AFEM). The AFEM is here used to model strong discontinuities independently of the underlying mesh. One noticeable advantage of the AFEM over other partition of unity methods is that it does not introduce additional global unknowns to represent cracks. Numerical 2D experiments illustrate the performance of the method and draw comparisons with the element deletion method (EDM), the phantom node method (PNM), the finite element method (FEM) and the embedded finite element method (EFEM). The h-convergence in the energy norm of the AFEM is studied for the first time and it is shown to outperform the aforementioned numerical methods when cracks are loaded in Mode I.

Error estimate of a decoupled numerical scheme for the Cahn–Hilliard–Stokes–Darcy system

Abstract We analyze a fully discrete finite element numerical scheme for the Cahn–Hilliard–Stokes–Darcy system that models two-phase flows in coupled free flow and porous media. To avoid a well-known difficulty associated with the coupling between the Cahn–Hilliard equation and the fluid motion, we make use of the operator-splitting in the numerical scheme, so that these two solvers are decoupled, which in turn would greatly improve the computational efficiency. The unique solvability and the energy stability have been proved in Chen et al. (2017, Uniquely solvable and energy stable decoupled numerical schemes for the Cahn–Hilliard–Stokes–Darcy system for two-phase flows in karstic geometry. Numer. Math., 137, 229–255). In this work, we carry out a detailed convergence analysis and error estimate for the fully discrete finite element scheme, so that the optimal rate convergence order is established in the energy norm, i.e., in the $\ell ^{\infty } (0, T; H^1) \cap \ell ^2 (0, T; H^2)$ norm for the phase variables, as well as in the $\ell ^{\infty } (0, T; H^1) \cap \ell ^2 (0, T; H^2)$ norm for the velocity variable. Such an energy norm error estimate leads to a cancelation of a nonlinear error term associated with the convection part, which turns out to be a key step to pass through the analysis. In addition, a discrete $\ell ^2 (0;T; H^3)$ bound of the numerical solution for the phase variables plays an important role in the error estimate, which is accomplished via a discrete version of Gagliardo–Nirenberg inequality in the finite element setting.

Sharp error estimates of a spectral Galerkin method for a diffusion-reaction equation with integral fractional Laplacian on a disk

We investigate a spectral Galerkin method for the two-dimensional fractional diffusion-reaction equations on a disk. We first prove regularity estimates of solutions in the weighted Sobolev space. Then we obtain optimal convergence orders of a spectral Galerkin method for the fractional diffusion-reaction equations in the L 2 L^2 and energy norm. We present numerical results to verify the theoretical analysis.

A residual based a posteriori error estimators for AFC schemes for convection-diffusion equations

In this work, we propose a residual-based a posteriori error estimator for algebraic flux-corrected (AFC) schemes for stationary convection-diffusion equations. A global upper bound is derived for the error in the energy norm for a general choice of the limiter, which defines the nonlinear stabilization term. In the diffusion-dominated regime, the estimator has the same convergence properties as the true error. A second approach is discussed, where the upper bound is derived in a posteriori way using the Streamline Upwind Petrov Galerkin (SUPG) estimator proposed in [20]. Numerical examples study the effectivity index and the adaptive grid refinement for two limiters in two dimensions.

Convergence and Supercloseness in a Balanced Norm of Finite Element Methods on Bakhvalov-Type Meshes for Reaction-Diffusion Problems

In convergence analysis of finite element methods for singularly perturbed reaction–diffusion problems, balanced norms have been successfully introduced to replace standard energy norms so that layers can be captured. In this article, we focus on the convergence analysis in a balanced norm on Bakhvalov-type rectangular meshes. In order to achieve our goal, a novel interpolation operator, which consists of a local $$L^2$$ projection operator and the Lagrange interpolation operator, is introduced for a convergence analysis of optimal order in the balanced norm. The analysis also depends on the stabilities of the $$L^2$$ projection and the characteristics of Bakhvalov-type meshes. Furthermore, we obtain a supercloseness result in the balanced norm, which appears in the literature for the first time. This result depends on another novel interpolant, which consists of the local $$L^2$$ projection operator, a vertices-edges-element operator and some corrections on the boundary.

Local Existence of Classical Solutions to Scalar Field Equation on Spatially Compact Spacetime as a Background

The goal of this paper is to prove the wellposedness of scalar field equation on spatially compact spacetime of Riemannian manifold. We construct the equation of motion from the Lagrangian of scalar field with non-minimal coupling, where the coupling interaction of the scalar field ϕ is proportional to the scalar curvature of the spacetime. The equation of motion has the form like non-linear wave equation. The next step is to prove local existence of solutions. We have show that both the k th linear energy and energy norm are bounded for the finite time with the initial data in H k+1 × H k. Finally, we prove the uniqueness and smoothness properties of the solution.

A Sequential Least Squares Method for Elliptic Equations in Non-Divergence Form

We develop a new least squares method for solving the second-order elliptic equations in non-divergence form. Two least-squares-type functionals are proposed for solving the equations in two steps. We first obtain a numerical approximation to the gradient in a piecewisely irrotational polynomial space. Then together with the numerical gradient, we seek a numerical solution of the primitive variable in continuous finite element space. The variational setting naturally provides a posteriori error which could be used in an adaptive refinement algorithm. The error estimates in $L^2$ norm and energy norms for both two unknowns are derived. By a series of numerical experiments, we verify the convergence rates and show the efficiency of the adaptive algorithm.

Efficient prediction of the effective nonlinear properties of porous material by FEM-Cluster based Analysis (FCA)

The present paper extends the FEM-Cluster based Analysis (FCA) method to the efficient prediction of effective properties of porous materials in the nonlinear range. Different from the traditional methods of homogenization or mean field theory, this new approach does not fill the pore with low modulus materials which causes numerical inaccuracy and convergence difficulties as reported in several literature. With no material filling in pores, the average strain theorem and the Hill–Mandel condition which is the basis of the Representative Volume Element (RVE) method for the prediction of effective properties of heterogeneous materials is almost impossible to implement. Furthermore, because of the lack of pore strain information, the compatibility condition of cluster strains cannot be applied in the incremental analysis of reduced order model in the nonlinear stage. In the present study we have overcome these difficulties by several approaches. First, it is found that the energy equivalence principle provides a more suitable basis for effective properties prediction of porous materials. Second, the self-equilibrium cluster stress field obtained from the interaction matrix and the cluster-based minimum complementary energy in FCA can be successfully applied to the incremental analysis of reduced order model in the nonlinear stage instead of the strain compatibility condition. However, due to the errors introduced by averaging stress and strain over each cluster, the interaction matrix becomes non-singular and the self-equilibrium properties of the cluster stress field generated from the interaction matrix become very poor. Fortunately, the most exciting discovery of the present study is that we have found a way to recover the singularity of the interaction matrix by minimizing its rank on the premise that the elastic energy norm of RVE remains unchanged within the given tolerance. Several numerical examples are presented to demonstrate the effectiveness and efficiency of the proposed approach.

English

We introduce and study various discontinuous Galerkin (DG) finite element approximations for a parabolic variational inequality associated with a general obstacle problem in ℝd (d = 2, 3). For the fully-discrete DG scheme, we employ a piecewise linear finite element space for spatial discretization, whereas the time discretization is carried out with the implicit backward Euler method. We present a unified error analysis for all well known symmetric and non-symmetric DG fully discrete schemes, and derive error estimate of optimal order $${\cal O}(h + \Delta t)$$ in an energy norm. Moreover, the analysis is performed without any assumptions on the speed of propagation of the free boundary and only the realistic regularity $${u_t} \in {{\cal L}^2}(0,T;{{\cal L}^2}(\Omega ))$$ is assumed. Further, we present some numerical experiments to illustrate the performance of the proposed methods.

Microstructure-informed reduced modes synthesized with Wang tiles and the Generalized Finite Element Method

A recently introduced representation by a set of Wang tiles—a generalization of the traditional Periodic Unit Cell-based approach—serves as a reduced geometrical model for materials with stochastic heterogeneous microstructure, enabling an efficient synthesis of microstructural realizations. To facilitate macroscopic analyses with a fully resolved microstructure generated with Wang tiles, we develop a reduced order modelling scheme utilizing pre-computed characteristic features of the tiles. In the offline phase, inspired by computational homogenization, we extract continuous fluctuation fields from the compressed microstructural representation as responses to generalized loading represented by the first- and second-order macroscopic gradients. In the online phase, using the ansatz of the generalized finite element method, we combine these fields with a coarse finite element discretization to create microstructure-informed reduced modes specific for a given macroscopic problem. Considering a two-dimensional scalar elliptic problem, we demonstrate that our scheme delivers less than 3% error in both the relative \(L_2\) and energy norms with only 0.01% of the unknowns when compared to the fully resolved problem. Accuracy can be further improved by locally refining the macroscopic discretization and/or employing more pre-computed fluctuation fields. Finally, unlike standard snapshot-based reduced-order approaches, our scheme handles significant changes in the macroscopic geometry or loading without the need for recalculating the offline phase, because the fluctuation fields are extracted without any prior knowledge of the macroscopic problem.

Solving three-dimensional interface problems with immersed finite elements: A-priori error analysis

Immersed finite element methods are designed to solve interface problems on interface-unfitted meshes. However, most of the study, especially analysis, is mainly limited to the two-dimension case. In this paper, we provide an a priori analysis for the trilinear immersed finite element method to solve three-dimensional elliptic interface problems on Cartesian grids consisting of cuboids. We establish the trace and inverse inequalities for trilinear IFE functions for interface elements with arbitrary interface-cutting configuration. Optimal a priori error estimates are rigorously proved in both energy and L2 norms, with the constant in the error bound independent of the interface location and its dependence on coefficient contrast explicitly specified. Numerical examples are provided not only to verify our theoretical results but also to demonstrate the applicability of this IFE method in tackling some real-world 3D interface models.

Analysis of the shifted boundary method for the Poisson problem in domains with corners

The shifted boundary method (SBM) is an approximate domain method for boundary value problems, in the broader class of unfitted/embedded/immersed methods. It has proven to be quite efficient in handling problems with complex geometries, ranging from Poisson to Darcy, from Navier-Stokes to elasticity and beyond. The key feature of the SBM is a shift in the location where Dirichlet boundary conditions are applied—from the true to a surrogate boundary—and an appropriate modification (again, a shift) of the value of the boundary conditions, in order to reduce the consistency error. In this paper we provide a sound analysis of the method in smooth domains and in domains with corners, highlighting the influence of geometry and distance between exact and surrogate boundaries upon the convergence rate. We consider the Poisson problem with Dirichlet boundary conditions as a model and we first detail a procedure to obtain the crucial shifting between the surrogate and the true boundaries. Next, we give a sufficient condition for the well-posedness and stability of the discrete problem. The behavior of the consistency error arising from shifting the boundary conditions is thoroughly analyzed, for smooth boundaries and for boundaries with corners and edges. The convergence rate is proven to be optimal in the energy norm, and is further enhanced in the L 2 L^2 -norm.

A quasi-optimal test norm for a DPG discretization of the convection-diffusion equation

In this work, we propose a new quasi-optimal test norm for a discontinuous Petrov-Galerkin (DPG) discretization of the ultra-weak formulation of the convection-diffusion equation. We prove theoretically that the proposed test norm leads to bounds between the target norm and the energy norm induced by the test norm which have favorable scalings with respect to the diffusion parameter when compared with existing results for other test norms from the literature. We conclude with numerical experiments to confirm our theoretical results.

Thermal flows in fractured porous media

We consider the thermal flow problem occuring in a fractured porous medium. The incompressible filtration flow in the porous matrix and the viscous flow in the fractures obey the Boussinesq approximation of Darcy-Forchheimer law and respectively, the Stokes system. They are coupled by the Saffman’s variant of the Beavers–Joseph condition. Existence and uniqueness properties are presented. The use of the energy norm in describing the Darcy-Forchheimer law proves to be appropriate. In the ε-periodic framework, we find the two-scale homogenized system which governs their asymptotic behaviours when ε → 0 and the Forchheimer effect vanishes. The limit problem is mainly a model of two coupled thermal flows, neither of them being incompressible.

A generalized finite element method for problems with sign-changing coefficients

Problems with sign-changing coefficients occur, for instance, in the study of transmission problems with metamaterials. In this work, we present and analyze a generalized finite element method in the spirit of the localized orthogonal decomposition, that is especially efficient when the negative and positive materials exhibit multiscale features. We derive optimal linear convergence in the energy norm independently of the potentially low regularity of the exact solution. Numerical experiments illustrate the theoretical convergence rates and show the applicability of the method for a large class of sign-changing diffusion problems.

Mixed Finite Element Method for Modified Poisson–Nernst–Planck/Navier–Stokes Equations

In this paper, a complete mixed finite element method is developed for a modified Poisson–Nernst–Planck/Navier–Stokes (PNP/NS) coupling system, where the original Poisson equation in PNP system is replaced by a fourth-order elliptic equation to more precisely account for electrostatic correlations in a simplified form of the Landau–Ginzburg-type continuum model. A stabilized mixed weak form is defined for each equation of the modified PNP/NS model in terms of primary variables and their corresponding vector-valued gradient variables, based on which a stable Stokes-pair mixed finite element is thus able to be utilized to discretize all solutions to the entire modified PNP/NS model in the framework of Stokes-type mixed finite element approximation. Semi- and fully discrete mixed finite element schemes are developed and are analyzed for the presented modified PNP/NS equations, and optimal convergence rates in energy norms are obtained for both schemes. Numerical experiments are carried out to validate all attained theoretical results.

A Nitsche extended finite element method for the biharmonic interface problem

In this paper, a Nitsche extended finite element method is presented to discretize the biharmonic interface problem with unfitted meshes. A new interface condition is proposed for the biharmonic interface problem, and the construction of the finite element space is based on the so-called modified Morley finite element for the interface elements and the Morley finite element for the others. By adding a stabilization procedure, we obtain the well-posedness for the discrete problem and prove an optimal a priori error estimate in the energy norm. It is shown that all results are uniform with respect to the mesh size, the material parameter quotient, and the position of the interface. Finally, numerical experiments are carried out to verify theoretical results.

Damage mechanism modelling of shield tunnel with longitudinal differential deformation based on elastoplastic damage model

Shield tunnels with great longitudinal differential deformations will adversely affect the service performance and optional safety of the metro system. Previous studies on the longitudinal mechanical behavior of shield tunnel are mostly carried out within the framework of elastic or elastic–plastic theory. However, assuming the concrete as elastoplastic body cannot properly reflect its nonlinear mechanical characteristics such as strain softening, stiffness degradation, etc., while damage or crack inevitably occurs when shield tunnel suffers large deformations. Therefore, the nonlinear damage characteristics of concrete material are considered in this paper, a novel positive/negative decomposition of stress tensor in energy norm is introduced herein to consider the asymmetric tensile/compressive material behavior of concrete, and a bi–scalar damage constitutive model of concrete is developed in turn. Then, to investigate the damage and degradation mechanisms of shield tunnel with differential deformation, the 3D discontinuous contact model is employed to develop the elaborate tunnel–soil numerical model. Results show that when shield tunnel suffers differential deformation, tensile damage dominates while compressive damage is minor, additional shear force and bending moment are induced, the ovality of tunnel cross section and serviceability also vary along the longitudinal direction. The damage and degradation of concrete material will reduce the tunnel integral stiffness and attenuate its ability to resist longitudinal and circumferential deformation. Besides, the segmental rebar is hard to yield, while the longitudinal coupling bolts on the lower half of tunnel segments with large deformation are inclined to yield. It should be noted that, the segmental rings near the inflection point of longitudinal deformation profile are most severely damaged, and exhibit the largest convergence deformation and lowest serviceability, where special attentions should be paid.