Bilinear Finite Element Research Articles

An analysis of the bilinear finite volume method for the singularly-perturbed convection-diffusion problems on Shishkin mesh

In this paper, we study the bilinear finite volume element method for solving the singularly perturbed convection-diffusion problem on the Shishkin mesh. We first prove that the finite volume element scheme is ϵ-uniformly stable. Then, based on new expression of the finite volume bilinear form and some detailed integral calculations, an ϵ-uniform error estimation is derived in the ϵ-weighted gradient norm, including the L2-norm. This error estimate is better than the known result. Moreover, we also give the L∞-error estimate near the boundary layer regions. At last, numerical experiments show the effectiveness of our method.

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.

A closed‐form multigrid smoothing factor for an additive Vanka‐type smoother applied to the Poisson equation

AbstractWe consider an additive Vanka‐type smoother for the Poisson equation discretized by the standard finite difference centered scheme. Using local Fourier analysis, we derive analytical formulas for the optimal smoothing factors for vertex‐wise and element‐wise Vanka smoothers. In one dimension the element‐wise Vanka smoother is equivalent to the scaled mass operator obtained from the linear finite element method and in two dimensions the element‐wise Vanka smoother is equivalent to the scaled mass operator discretized by bilinear finite element method plus a scaled identity operator. Based on these findings, the mass matrix obtained from finite element method can be used as a smoother for the Poisson equation, and the resulting mass‐based relaxation scheme yields small smoothing factors in one, two, and three dimensions, while avoiding the need to compute an inverse of a matrix. Our analysis may help better understand the smoothing properties of additive Vanka approaches and develop fast solvers for numerical solutions of other partial differential equations.

An analysis of the isoparametric bilinear finite volume element method by applying the Simpson rule to quadrilateral meshes

<abstract><p>In this work, we construct and study a special isoparametric bilinear finite volume element scheme for solving anisotropic diffusion problems on general convex quadrilateral meshes. The new scheme is obtained by employing the Simpson rule to approximate the line integrals in the classical isoparametric bilinear finite volume element method. By using the cell analysis approach, we suggest a sufficient condition to ensure the coercivity of the new scheme. The sufficient condition has an analytic expression, which only involves the anisotropic diffusion tensor and the geometry of quadrilateral mesh. This yields that for any diffusion tensor and quadrilateral mesh, we can directly judge whether this sufficient condition is satisfied. Specifically, this condition covers the traditional $ h^{1+\gamma} $-parallelogram and some trapezoidal meshes with any full anisotropic diffusion tensor. An optimal $ H^1 $ error estimate of the proposed scheme is also obtained for a quasi-parallelogram mesh. The theoretical results are verified by some numerical experiments.</p></abstract>

Superconvergence analysis of anisotropic finite element method for the time fractional substantial diffusion equation with smooth and nonsmooth solutions

In this work, anisotropic bilinear finite element and second‐order temporal approximation are adopted to establish a fully discrete scheme for the fractional substantial diffusion equation with variable coefficient. We prove the proposed discrete scheme is unconditionally stable in the senses of ‐norm and ‐norm. By introducing a new projection, the optimal convergence error in ‐norm and the superclose property in ‐norm are obtained under the condition of , where is the exact solution of the problem. Through interpolation post processing technique, we arrive at the global superconvergence of the interpolation. Furthermore, an improved algorithm is proposed to solve the problem with nonsmooth solution. Finally, numerical tests are given to illustrate the validity and efficiency of our theoretical analysis.

Novel mass-based multigrid relaxation schemes for the Stokes equations

In this work, we propose three novel block-structured multigrid relaxation schemes based on distributive relaxation, Braess–Sarazin relaxation, and Uzawa relaxation, for solving the Stokes equations discretized by the marker-and-cell scheme. The mass matrix obtained from the bilinear finite element method is directly used to approximate the inverse of scalar Laplacian operator in the relaxation schemes. Using local Fourier analysis, we theoretically derive optimal smoothing factors for the resulting three relaxation schemes. Specifically, mass-based distributive relaxation, mass-based Braess–Sarazin relaxation, and mass-based σ-Uzawa relaxation have optimal smoothing factor 13, 13 and 13, respectively. These smoothing factors are smaller than those in our earlier work [1], where weighted Jacobi iteration is used for inventing Laplacian involved in the Stokes equations. The mass-based relaxation schemes do not cost more than the original ones using the Jacobi iteration. Another superiority of mass-based relaxation is that there is no need to compute the inverse of a matrix. These new relaxation schemes are appealing.

On the convergence and superconvergence for a class of two‐dimensional time fractional reaction‐subdiffusion equations

AbstractThis paper analyzes a class of two‐dimensional (2‐D) time fractional reaction‐subdiffusion equations with variable coefficients. The high‐order L2‐1σ time‐stepping scheme on graded meshes is presented to deal with the weak singularity at the initial time t = 0, and the bilinear finite element method (FEM) on anisotropic meshes is used for spatial discretization. Using the modified discrete fractional Grönwall inequality, and combining the interpolation operator and the projection operator, the L2‐norm error estimation and H1‐norm superclose results are rigorously proved. The superconvergence result in the H1‐norm is derived by applying the interpolation postprocessing technique. Finally, numerical examples are presented to verify the validation of our theoretical analysis.

Superconvergence analysis for nonlinear viscoelastic wave equation with strong damping

ABSTRACT In this research, with help of the bilinear finite element, superconvergence analysis is proposed for nonlinear viscoelastic wave equation with strong damping. Firstly, a temporal discrete approximate scheme is established, and assisted by the method of first hypothesis and then proof, the temporal errors are given. Secondly, a new linearized second-order fully discrete scheme is presented, by using the technique of Interpolation and Ritz-prejection combination in the error estimations and analysis process, rigorous proofs are provided for the unconditional superclose estimates in -norm with order , here h and τ denote mesh size and time step, respectively. Finally, recurring to numerical examples, the correctness of the theoretical analysis is further demonstrated.

Discrete strong extremum principles for finite element solutions of diffusion problems with nonlinear corrections

A nonlinear correction technique for finite element methods to anisotropic diffusion problems on general triangular and quadrilateral meshes are introduced. The classic linear or bi-linear finite element methods are modified, and then the resulting schemes satisfy the discrete strong extremum principle unconditionally, which means that it is unnecessary to impose the well-known restrictions on diffusion coefficients and geometry of mesh-cell (e.g. “acute angle” condition). Convergence rate for smooth and piecewise smooth solutions and the discrete extremum principle property are verified by numerical examples.

DEFORMATION TRANSVERSE SHEAR BENDING STATE OF A THIN PLATE LAYER OF AN ANISOTROPIC GEOLOGICAL MEDIUM FROM THE ACTION OF CONCENTRATED ENERGY IMPULSES

A method is proposed for study the structural stability of the deformation state of structural blocks of the earth's crust, approximated in the form of plate layers of the geological medium when transverse shear bending from the action of concentrated energy impulses. Advances here are carried out in the two directions. First, in contrast to the previous article, the physical and mechanical model of the geological medium is endowed with anisotropic properties, which makes it possible to increase the adequacy of the obtained numerical results to the specifics of the real problem. Secondly, instead of the simplest bilinear 4-node finite elements, the special spectral non-algebraic 8-node finite iso-parametric finite elements are used, the use of which significantly increases both the accuracy of calculations and their reliability in the sense of ensuring the robustness of calculations for relatively small values of the plate thickness. It should be noted that the Finite Element Method uses exclusively only algebraic finite elements (power polynomials in the h-version and orthogonal polynomials in the p-version). It is known from approximation theory that the use of spectral non-algebraic approximations improves the quality of approximations. Therefore, their introduction into the structure of finite element calculations can improve the quality of modeling in the study of the strain-stress-state (SSS) of the geological medium. A structural block (SB) is understood as a plate layer with plan dimensions exceeding the thickness by more than 10 times. The identification of hazardous zones in the rock massive due to stress concentration is complemented by the development of mechanical, mathematical and computational tools for modeling the curvature of the earth's crust during bending based on the classical theory of Kirchhoff and refined Reissner-Mindlin theory. Test calculations have shown that the accuracy of the calculation and the quality of geometric modeling of fragments of an anisotropic geological environment based on the refined 8-node spectral finite element is significantly better than for the 8-node algebraic finite

A nonuniform anisotropic FEM for elliptic boundary layer optimal control problems

In this paper, an anisotropic bilinear finite element method is constructed for the elliptic boundary layer optimal control problems. Supercloseness properties of the numerical state and numerical adjoint state in a \begin{document}$ \epsilon $\end{document} -norm are established on anisotropic meshes. Moreover, an interpolation type post-processed solution is shown to be superconvergent of order \begin{document}$ O(N^{-2}) $\end{document} , where the total number of nodes is of \begin{document}$ O(N^2) $\end{document} . Finally, numerical results are provided to verify the theoretical analysis.

A linearity preserving nodal variation limiting algorithm for continuous Galerkin discretization of ideal MHD equations

In this work, a stabilized continuous Galerkin (CG) method for magnetohydrodynamics (MHD) is presented. Ideal, compressible inviscid MHD equations are discretized in space on unstructured meshes using piecewise linear or bilinear finite element bases to get a semi-discrete scheme. Stabilization is then introduced to the semi-discrete method in a strategy that follows the algebraic flux correction paradigm. This involves adding some artificial diffusion to the high order, semi-discrete method and mass lumping in the time derivative term. The result is a low order method that provides local extremum diminishing properties for hyperbolic systems. The difference between the low order method and the high order method is scaled element-wise using a limiter and added to the low order scheme. The limiter is solution dependent and computed via an iterative linearity preserving nodal variation limiting strategy. The stabilization also involves an optional consistent background high order dissipation that reduces phase errors. The resulting stabilized scheme is a semi-discrete method that can be applied to inviscid shock MHD problems and may be even extended to resistive and viscous MHD problems. To satisfy the divergence free constraint of the MHD equations, we add parabolic divergence cleaning to the system. Various time integration methods can be used to discretize the scheme in time. We demonstrate the robustness of the scheme by solving several shock MHD problems.

Superconvergence analysis of a two-grid method for nonlinear hyperbolic equations

In this paper, the superconvergence analysis of a second-order fully discrete scheme with two-grid method (TGM) is studied for the two-dimensional nonlinear hyperbolic equations by the bilinear finite element. The existence and uniqueness of the solution for the scheme are proved rigorously. By use of the combination technique of the interpolation and Ritz projection, the superclose results in H1-norm are obtained, and the global superconvergence is derived through the interpolated postprocessing approach. Finally, some numerical results are provided to verify the theoretical predictions and show the efficiency of the proposed TGM.

Local ultraconvergence of linear and bilinear finite element method for second order elliptic problems

In this article, local ultraconvergence in gradient of the bilinear and linear finite element solutions to two-dimensional second order elliptic problems has been investigated. Special interpolation postprocessing algorithms of numerical solutions by Richardson extrapolation have been developed. The local symmetry theory and estimates of discrete Green functions are used in the analysis. Numerical experiments are provided to confirm the theoretical findings.

M6 USING PRS SCORES TO CLARIFY NOSOLOGICAL STRUCTURE OF ADHD AND OVERLAPPING EMOTION-RELATED TRAIT MEASURES

In this work, a stabilized continuous Galerkin (CG) method for magnetohydrodynamics (MHD) is presented. Ideal, compressible inviscid MHD equations are discretized in space on unstructured meshes using piecewise linear or bilinear finite element bases to get a semi-discrete scheme. Stabilization is then introduced to the semi-discrete method in a strategy that follows the algebraic flux correction paradigm. This involves adding some artificial diffusion to the high order, semi-discrete method and mass lumping in the time derivative term. The result is a low order method that provides local extremum diminishing properties for hyperbolic systems. The difference between the low order method and the high order method is scaled element-wise using a limiter and added to the low order scheme. The limiter is solution dependent and computed via an iterative linearity preserving nodal variation limiting strategy. The stabilization also involves an optional consistent background high order dissipation that reduces phase errors. The resulting stabilized scheme is a semi-discrete method that can be applied to inviscid shock MHD problems and may be even extended to resistive and viscous MHD problems. To satisfy the divergence free constraint of the MHD equations, we add parabolic divergence cleaning to the system. Various time integration methods can be used to discretize the scheme in time. We demonstrate the robustness of the scheme by solving several shock MHD problems.

Superconvergence analysis of the mixed finite element method for the Rosenau equation

In this paper, an implicit Crank-Nicolson (CN) formula of the mixed finite element method (FEM) is developed with the bilinear finite element for nonlinear fourth-order Rosenau equation. The stability, existence and uniqueness of the approximate solution of this scheme are proved. Then, unconditional superclose and superconvergence estimates of order O(h2+τ2) in H1-norm are derived by the combination technique of interpolation and projection. Finally, some numerical results confirm our theoretical analysis. Here and later h and τ denote the mesh size and the time step, respectively.

Evaluation of ASTM D5764 Dowel Connection Tests for Laminated Veneer Bamboo (LVB)

Abstract The ASTM D5764 standard, Standard Test Method for Evaluating Dowel-Bearing Strength of Wood and Wood-Based Products, for testing dowel connections provides a procedure for measuring the dowel bearing strength of wood and wood-based products. Laminated veneer bamboo (LVB) is a new building product that is employed in similar sizes and applications as dimensional lumber. Being new, more research is needed to understand the key factors and fundamental failure mechanisms that occur in LVB dowel connections to help ensure safe standards for further LVB product adoption and design. This study develops three-dimensional bilinear finite element models for half- and full-hole specimens in accordance with ASTM D5764 when loaded in compression parallel to the grain. The models simulate LVB fracture initiation due to shear stresses in the dowel joint by incorporating frictional stresses in the contact region between a steel bolt and LVB. The model also predicts displacement at failure, which is validated through comparison with experimental results: the material fails at 1 and 1.18 mm displacement loading parallel to the grain for half- and full-hole specimens, respectively. It is found that, despite the higher load-bearing capacity (strength) of the half-hole specimen, both specimens fail at approximately the same displacement because of in-plane shear stresses. This article clarifies the complex interactive state of in-plane shear, tension perpendicular to the grain, and compression parallel-to-grain stresses using the Tsai–Wu failure criterion in the critical zone beneath the bolt hole for half- and full-hole specimens. These findings suggest that care should be taken to select a test method that captures the performance of LVB dowel joints because of different failure mechanisms that occur for full- and half-hole specimens.

Superconvergence analysis of finite element method for Poisson–Nernst–Planck equations

This article concerns with the superconvergence analysis of bilinear finite element method (FEM) for nonlinear Poisson–Nernst–Planck (PNP) equations. By employing high accuracy integral identities together with mean value technique, the superclose estimates in H1‐norm are derived for the semi‐discrete and the backward Euler fully‐discrete schemes, which improve the suboptimal error estimate in L2‐norm in the previous literature. Furthermore, the global superconvergence results in H1‐norm are obtained through interpolation postprocessing approach. Finally, a numerical example is provided to confirm the theoretical analysis.

On a finite element method for measure-valued optimal control problems governed by the 1D generalized wave equation

The paper deals with the optimal control problems governed by the 1D wave equation with variable coefficients and the control spaces of either measure-valued functions Lw⁎2(I,M(Ω)) or vector measures M(Ω,L2(I)). Bilinear finite element discretizations are constructed and their stability and error analysis is accomplished.

A novel approach of high accuracy analysis of anisotropic bilinear finite element for time-fractional diffusion equations with variable coefficient

In this paper, by using bilinear finite element and L1 approximation, a fully-discrete scheme is established for time fractional diffusion equation with variable coefficient on anisotropic meshes. Unconditionally stable analysis of the proposed scheme are presented in both L2-norm and H1-norm. Moreover, convergence, superclose and superconvergence results are derived by combining interpolation with projection, which is the key technique for the numerical analysis. Specifically, by defining a novel projection operator, the error estimate between the projection and the exact solution is obtained on anisotropic meshes. Furthermore, high accuracy analysis on interpolation of bilinear finite element and projection is gained by means of some known results about the interpolation and mean value technique. Based on the related results about projection and interpolation, the optimal error estimate in L2-norm and superclose of interpolation in H1-norm are deduced by skillfully dealing with fractional derivative. At the same time, the global superconvergence is presented by employing interpolation postprocessing operator. Finally, numerical results are provided to demonstrate the validity of the theoretical analysis.