Piecewise Quadratic Finite Element Research Articles

A Nitsche mixed extended finite element method for the biharmonic interface problem

In this paper, a Nitsche mixed extended finite element method based on Ciarlet–Raviart formulation is proposed to discretize the biharmonic interface problem with interface unfitted meshes. We present a discrete mixed approximate formulation based on the piecewise quadratic continuous finite element space. By adding a stabilization procedure, we get the discrete inf–sup condition and prove a suboptimal a priori error estimate for the discrete problem. If the solution of the dual problem satisfies the additional regularity condition, then an optimal a priori error estimate for stream variable under piecewise H1-norm is obtained. It is shown that all results are uniform with respect to the mesh size, and the location of the interface relative to the mesh. Finally, numerical experiments are carried out to demonstrate our theoretical results.

Fully Discrete Finite Element Approximation of the MHD Flow

Abstract In this work, we consider a fully discrete finite element approximation of the 3D incompressible magnetohydrodynamic system. The velocity and magnetic field are approximated both by piecewise quadratic finite elements, while the pressure is approximated by piecewise linear finite elements. The time discretization is based on the Crank–Nicolson scheme for the linear terms in the model and the explicit Adams–Bashforth for the nonlinear terms. We establish the optimal error estimates of both the approximate velocity and magnetic field in H 1 \mathbf{H}^{1} -norm and of the approximate pressure in L 2 L^{2} -norm. In order to achieve the optimal L 2 L^{2} -norm error estimates of both the approximate velocity and magnetic field, we shall make use of a special negative norm technique.

Optimal quadratic element on rectangular grids for $$H^1$$ problems

In this paper, a piecewise quadratic finite element method on rectangular grids for \(H^1\) problems is presented. The proposed method can be viewed as a reduced rectangular Morley element. For the source problem, the convergence rate of this scheme is proved to be \(O(h^2)\) in the energy norm on uniform grids over a convex domain. A lower bound of the \(L^2\)-norm error is also proved, which makes the capacity of this scheme more clear. For the eigenvalue problem, the computed eigenvalues by this element are shown to be the lower bounds of the exact ones. Some numerical results are presented to verify the theoretical findings.

A quadratic finite element wavelet Riesz basis

In this paper, continuous piecewise quadratic finite element wavelets are constructed on general polygons in [Formula: see text]. The wavelets are stable in [Formula: see text] for [Formula: see text] and have two vanishing moments. Each wavelet is a linear combination of 11 or 13 nodal basis functions. Numerically computed condition numbers for [Formula: see text] are provided for the unit square.

A Posteriori Error Estimation for Planar Linear Elasticity by Stress Reconstruction

Abstract The nonconforming triangular piecewise quadratic finite element space by Fortin and Soulie can be used for the displacement approximation and its combination with discontinuous piecewise linear pressure elements is known to constitute a stable combination for incompressible linear elasticity computations. In this contribution, we extend the stress reconstruction procedure and resulting guaranteed a posteriori error estimator developed by Ainsworth, Allendes, Barrenechea and Rankin [2] and by Kim [18] to linear elasticity. In order to get a guaranteed reliability bound with respect to the energy norm involving only known constants, two modifications are carried out: (i) the stress reconstruction in next-to-lowest order Raviart–Thomas spaces is modified in such a way that its anti-symmetric part vanishes in average on each element; (ii) the auxiliary conforming approximation is constructed under the constraint that its divergence coincides with the one for the nonconforming approximation. An important aspect of our construction is that all results hold uniformly in the incompressible limit. Global efficiency is also shown and the effectiveness is illustrated by adaptive computations involving different Lamé parameters including the incompressible limit case.

A continuous/discontinuous Galerkin method and a priori error estimates for the biharmonic problem on surfaces

We present a continuous/discontinuous Galerkin method for approximating solutions to a fourth order elliptic PDE on a surface embedded in $\mathbb{R}^3$. A priori error estimates, taking both the approximation of the surface and the approximation of surface differential operators into account, are proven in a discrete energy norm and in $L^2$-norm. This can be seen as an extension of the formalism and method originally used by Dziuk [14] for approximating solutions to the Laplace-Beltrami problem, and within this setting this is the first analysis of a surface finite element method formulated using higher order surface differential operators. Using a polygonal approximation $\Gamma_h$ of an implicitly defined surface $\Gamma$ we employ continuous piecewise quadratic finite elements to approximate solutions to the biharmonic equation on $\Gamma$. Numerical examples on the sphere and on the torus confirm the convergence rate implied by our estimates.

Functional Parallel Factor Analysis for Functions of One- and Two-dimensional Arguments

Parallel factor analysis (PARAFAC) is a useful multivariate method for decomposing three-way data that consist of three different types of entities simultaneously. This method estimates trilinear components, each of which is a low-dimensional representation of a set of entities, often called a mode, to explain the maximum variance of the data. Functional PARAFAC permits the entities in different modes to be smooth functions or curves, varying over a continuum, rather than a collection of unconnected responses. The existing functional PARAFAC methods handle functions of a one-dimensional argument (e.g., time) only. In this paper, we propose a new extension of functional PARAFAC for handling three-way data whose responses are sequenced along both a two-dimensional domain (e.g., a plane with x- and y-axis coordinates) and a one-dimensional argument. Technically, the proposed method combines PARAFAC with basis function expansion approximations, using a set of piecewise quadratic finite element basis functions for estimating two-dimensional smooth functions and a set of one-dimensional basis functions for estimating one-dimensional smooth functions. In a simulation study, the proposed method appeared to outperform the conventional PARAFAC. We apply the method to EEG data to demonstrate its empirical usefulness.

Linear differential operators on bivariate spline spaces and spline vector fields

We consider the application of standard differentiation operators to spline spaces and spline vector fields defined on triangulations in the plane. In particular, we explore the use of Bernstein–Bezier techniques for answering questions such as: What are the images or the kernels, and their dimensions, of partial derivative, gradient, divergence, curl, or Laplace, operators. We also describe a particular continuous piecewise quadratic finite element whose nodal parameters are function and divergence (or curl) values.

A simple finite element method for boundary value problems with a Riemann–Liouville derivative

We consider a boundary value problem involving a Riemann–Liouville fractional derivative of order α∈(3/2,2) on the unit interval (0,1). The standard Galerkin finite element approximation converges slowly due to the presence of singularity term xα−1 in the solution representation. In this work, we develop a simple technique, by transforming it into a second-order two-point boundary value problem with nonlocal low order terms, whose solution can reconstruct directly the solution to the original problem. The stability of the variational formulation, and the optimal regularity pickup of the solution are analyzed. A novel Galerkin finite element method with piecewise linear or quadratic finite elements is developed, and L2(D) error estimates are provided. The approach is then applied to the corresponding fractional Sturm–Liouville problem, and error estimates of the eigenvalue approximations are given. Extensive numerical results fully confirm our theoretical study.

A generalized multiscale finite element method for the Brinkman equation

In this paper we consider the numerical upscaling of the Brinkman equation in the presence of high-contrast permeability fields. We develop and analyze a robust and efficient Generalized Multiscale Finite Element Method (GMsFEM) for the Brinkman model in two dimensions. In the fine grid, we use mixed finite element method with the velocity and pressure being continuous piecewise quadratic and piecewise constant finite element spaces, respectively. Using the GMsFEM framework we construct suitable coarse-scale spaces for the velocity and pressure that yield a robust mixed GMsFEM. We develop a novel approach to construct a coarse approximation for the velocity snapshot space and a robust small offline space for the velocity space. The stability of the mixed GMsFEM and a priori error estimates are derived. A variety of two-dimensional numerical examples are presented to illustrate the effectiveness of the algorithm.

An interior penalty method for distributed optimal control problems governed by the biharmonic operator

In this paper, a C0 interior penalty method has been proposed and analyzed for distributed optimal control problems governed by the biharmonic operator. The state and adjoint variables are discretized using continuous piecewise quadratic finite elements while the control variable is discretized using piecewise constant approximations. A priori and a posteriori error estimates are derived for the state, adjoint and control variables under minimal regularity assumptions. Numerical results justify the theoretical results obtained. The a posteriori error estimators are useful in adaptive finite element approximation and the numerical results indicate that the sharp error estimators work efficiently in guiding the mesh refinement.

An adaptation of Nitscheʼs method to the Tresca friction problem

We propose a simple adaptation to the Tresca friction case of the Nitsche-based finite element method introduced previously for frictionless unilateral contact. Both cases of unilateral and bilateral contact with friction are taken into account, with emphasis on frictional unilateral contact for the numerical analysis. We manage to prove theoretically the fully optimal convergence rate of the method in the H1(Ω)-norm which is O(h12+ν) when the solution lies in H32+ν(Ω), 0<ν⩽k−1/2, in two dimensions and three dimensions, for Lagrange piecewise linear (k=1) and quadratic (k=2) finite elements. No additional assumption on the friction set is needed to obtain this proof.

Spatial Spline Regression Models

SummaryWe describe a model for the analysis of data distributed over irregularly shaped spatial domains with complex boundaries, strong concavities and interior holes. Adopting an approach that is typical of functional data analysis, we propose a spatial spline regression model that is computationally efficient, allows for spatially distributed covariate information and can impose various conditions over the boundaries of the domain. Accurate surface estimation is achieved by the use of piecewise linear and quadratic finite elements.

Improving mesh quality and finite element solution accuracy by GETMe smoothing in solving the Poisson equation

Mesh quality plays an essential role in finite element applications, since it affects the efficiency of the simulation with respect to solution accuracy and computational effort. Therefore, mesh smoothing techniques are often applied for improving mesh quality while preserving mesh topology. One of these methods is the recently proposed geometric element transformation method (GETMe), which is based on regularizing element transformations. It will be shown numerically that this smoothing method is particularly suitable, from an applicational point of view, since it leads to a significant reduction of discretization errors within the first few smoothing steps requiring only little computational effort. Furthermore, due to reduced condition numbers of the stiffness matrices the performance of iterative solvers of the resulting finite element equations is improved. This is demonstrated for the Poisson equation with a number of meshes of different complexity and type as well as for piecewise linear and quadratic finite element basis functions. Results are compared to those obtained by two variants of Laplacian smoothing and a state of the art global optimization-based approach.

A Characteristic Galerkin Method for Eddy-Current Field Analysis in High-Speed Rotating Solid Conductors

Standard Galerkin finite element method (FEM) performs poorly when simulating convection dominated problems using the Eulerian formulation. A numerical method for analyzing transient eddy-current magnetic field in high-speed rotating solid conductors using characteristic Galerkin finite element method (CGFEM) is presented. The convection-diffusion equation is discretized implicitly in time domain via the method of characteristics and in space via piecewise quadratic finite elements. An example with extremely large Peclet number is solved by using the CGFEM to validate its effectiveness. TEAM workshop problem 30 A which is a practical case with high-speed rotating conductors is studied to showcase the efficiency and accuracy of the proposed method.

An active strain electromechanical model for cardiac tissue.

We propose a finite element approximation of a system of partial differential equations describing the coupling between the propagation of electrical potential and large deformations of the cardiac tissue. The underlying mathematical model is based on the active strain assumption, in which it is assumed that there is a multiplicative decomposition of the deformation tensor into a passive and active part holds, the latter carrying the information of the electrical potential propagation and anisotropy of the cardiac tissue into the equations of either incompressible or compressible nonlinear elasticity, governing the mechanical response of the biological material. In addition, by changing from a Eulerian to a Lagrangian configuration, the bidomain or monodomain equations modeling the evolution of the electrical propagation exhibit a nonlinear diffusion term. Piecewise quadratic finite elements are employed to approximate the displacements field, whereas for pressure, electrical potentials and ionic variables are approximated by piecewise linear elements. Various numerical tests performed with a parallel finite element code illustrate that the proposed model can capture some important features of the electromechanical coupling and show that our numerical scheme is efficient and accurate.

Realistic computable error bounds for three dimensional finite element analyses in linear elasticity

We obtain a computable estimator for the energy norm of the error in piecewise affine and piecewise quadratic finite element approximations of linear elasticity in three dimensions. We show that the estimator provides guaranteed upper bounds on the energy norm of the error as well as (up to a constant and data oscillation terms) local lower bounds.

A method of computer assisted proof for nonlinear two-point boundary value problems using higher order finite elements

Present authors have presented with Takayuki Kubo at University of Tsukuba a method of a computer assisted proof for the existence and uniqueness of solutions to two-point boundary value problems of nonlinear ordinary differential equations in the paper submitted for NOLTA, IEICE. This method uses piecewise linear finite element base functions and sometimes requires fine mesh. To overcome this difficulty, in this paper, an improved method is presented for the norm estimation of the residual to the operator equation. In this refined formulation, piecewise quadratic finite element base functions are used. A kind of the residual technique works sophisticatedly well. It is stated that the estimation of the residual can be expected smaller than that of the previous method. Finally, four examples are presented. Each result demonstrates that a remarkable improvement is achieved in accuracy of the guaranteed error estimation.

A numerical method for pricing spread options on LIBOR rates with a PDE model

In this paper we present a new numerical method for solving a Black–Scholes type of model for pricing a class of interest rate derivatives: spread options on LIBOR rates. The interest rates are assumed to follow the recently introduced LIBOR Market Model. The Feynman–Kac theorem provides a PDE model for the spread option pricing problem which is initially posed in an unbounded domain. After a localization procedure and the consideration of appropriate boundary conditions in a bounded domain, we propose a Crank–Nicholson characteristic time discretization scheme combined with a Lagrange piecewise quadratic finite element for the spatial discretization. In order to illustrate the performance of the PDE model and the numerical methods, we present a real example of spread option pricing.

A comparison of solvers for linear complementarity problems arising from large-scale masonry structures

We compare the numerical performance of several methods for solving the discrete contact problem arising from the finite element discretisation of elastic systems with numerous contact points. The problem is formulated as a variational inequality and discretised using piecewise quadratic finite elements on a triangulation of the domain. At the discrete level, the variational inequality is reformulated as a classical linear complementarity system. We compare several state-of-art algorithms that have been advocated for such problems. Computational tests illustrate the use of these methods for a large collection of elastic bodies, such as a simplified bidimensional wall made of bricks or stone blocks, deformed under volume and surface forces.