Piecewise Linear Finite Elements In Space Research Articles

A monotone finite volume element scheme for diffusion equations on arbitrary polygonal grids

We develop a monotone finite volume element scheme for the diffusion problem on arbitrary polygonal grids inspired by the scheme in the literature ([35]). The main contributions of this paper include three aspects. Firstly, some auxiliary lines or points are introduced for any polygonal element so that it can be partitioned into two or more triangles, and one novel type of its control volume is concomitantly designed. In this paper, we mainly discuss quadrilateral grids. Secondly, some piecewise linear finite element spaces are imported for the trial spaces. Fictitious elements, continuous extensions and the nonlinear two-point flux idea can all be inherited from the triangular case. Therefore, the monotonicity of the scheme is easily proved only by the assembling of some triangles' element stiff matrices. Thirdly, the agreeable adaptability for distorted grids is completely borned. The strict convexity restriction on the grids can also be removed, the concave or severely distorted polygonal (such as quadrilateral) grids are transformed as the convex and general regular triangles. Finally, numerical results confirm its monotonicity, accuracy and stability.

Higher Order Finite Element Methods for Some One-dimensional Boundary Value Problems

In this paper, third-order compact and fourth-order finite element methods (FEMs) based on simple modifications of traditional FEMs are proposed for solving one-dimensional Sturm-Liouville boundary value problems (BVPs). The key idea is based on interpolation error estimates. A simple posterior error analysis of the original piecewise linear finite element space leads to a third-order accurate solution in the L2 norm, second-order in the H1, and the energy norm. The novel idea is also applied to obtain a fourth-order FEM based on the quadratic finite element space. The basis functions in the new fourth-order FEM are more compact compared with that of the classic cubic basis functions. Numerical examples presented in this paper have confirmed the convergence order and analysis. A generalization to a class of nonlinear two-point BVPs is also discussed and tested.

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.

Fully discrete a posteriori error estimates for parabolic integro-differential equations using the two-step backward differentiation formula

In this article, finite element a posteriori error estimates for the linear parabolic integro-differential equation using the two-step backward time descretization formula are explored. For space discretization, we use piecewise linear finite element spaces. The Ritz–Volterra reconstruction operator is used as a raw ingredient to obtain the optimal convergence in space. Further, a novel quadratic space–time reconstruction operator, namely BDF2 operator, is introduced to achieve second-order accuracy in time. Numerical results demonstrate the theoretical findings.

A Compact FEM Implementation for Parabolic Integro-Differential Equations in 2D

Although two-dimensional (2D) parabolic integro-differential equations (PIDEs) arise in many physical contexts, there is no generally available software that is able to solve them numerically. To remedy this situation, in this article, we provide a compact implementation for solving 2D PIDEs using the finite element method (FEM) on unstructured grids. Piecewise linear finite element spaces on triangles are used for the space discretization, whereas the time discretization is based on the backward-Euler and the Crank–Nicolson methods. The quadrature rules for discretizing the Volterra integral term are chosen so as to be consistent with the time-stepping schemes; a more efficient version of the implementation that uses a vectorization technique in the assembly process is also presented. The compactness of the approach is demonstrated using the software Matrix Laboratory (MATLAB). The efficiency is demonstrated via a numerical example on an L-shaped domain, for which a comparison is possible against the commercially available finite element software COMSOL Multiphysics. Moreover, further consideration indicates that COMSOL Multiphysics cannot be directly applied to 2D PIDEs containing more complex kernels in the Volterra integral term, whereas our method can. Consequently, the subroutines we present constitute a valuable open and validated resource for solving more general 2D PIDEs.

A Posteriori Error Estimates for Fully Discrete Finite Element Method for Generalized Diffusion Equation with Delay.

In this paper, we derive several a posteriori error estimators for generalized diffusion equation with delay in a convex polygonal domain. The Crank–Nicolson method for time discretization is used and a continuous, piecewise linear finite element space is employed for the space discretization. The a posteriori error estimators corresponding to space discretization are derived by using the interpolation estimates. Two different continuous, piecewise quadratic reconstructions are used to obtain the error due to the time discretization. To estimate the error in the approximation of the delay term, linear approximations of the delay term are used in a crucial way. As a consequence, a posteriori upper and lower error bounds for fully discrete approximation are derived for the first time. In particular, long-time a posteriori error estimates are obtained for stable systems. Numerical experiments are presented which confirm our theoretical results.

Finite element methods for non-fourier thermal wave model of bio heat transfer with an interface

We propose a fitted finite element method for non-Fourier bio heat transfer model in multi-layered media. Specifically, we employ the Maxwell–Cattaneo equation on the physical media that have a heterogeneous conductivity. Well-posedness of the model interface problem is established. A continuous piecewise linear finite element space is employed for the spatially semidiscrete approximation and the temporal discretization is based on backward scheme. Optimal order error estimates for both semidiscrete and fully discrete schemes are proved in $$L^{\infty }(H^1)$$ norm. Finally, we give numerical examples to verify our theoretical results. The new results and finite element schemes can be applied in the fields of engineering, medicine, and biotechnology.

Conforming and discontinuous Galerkin FEM in space for solving parabolic obstacle problem

In this article, we propose and analyze conforming and discontinuous Galerkin (DG) finite element methods for numerical approximation of the solution of the parabolic variational inequality associated with a general obstacle in Rd(d=2,3). For fully discrete conforming method, we use globally continuous and piecewise linear finite element space. Whereas for the fully-discrete DG scheme, we employ piecewise linear finite element space for spatial discretization. The time discretization has been done by using the implicit backward Euler method. We present the error analysis for the conforming and the DG fully discrete schemes and derive an error estimate of optimal order O(h+Δt) in a certain energy norm defined precisely in the article. The analysis is performed without any assumptions on the speed of propagation of the free boundary but only assumes the pragmatic regularity that ut∈L2(0,T;L2(Ω)). The obstacle constraints are incorporated at the Lagrange nodes of the triangular mesh and the analysis exploits the Lagrange interpolation. We present some numerical experiment to illustrate the performance of the proposed methods.

Space–time discontinuous Galerkin methods for the ε-dependent stochastic Allen–Cahn equation with mild noise

Abstract We consider the $\varepsilon $-dependent stochastic Allen–Cahn equation with mild space–time noise posed on a bounded domain of $\mathbb{R}^2$. The positive parameter $\varepsilon $ is a measure for the inner layers width that are generated during evolution. This equation, when the noise depends only on time, has been proposed by Funaki (1999, Singular limit for stochastic reaction–diffusion equation and generation of random interfaces. Acta Math. Sin., 15, 407–438). The noise, although smooth, becomes white on the sharp interface limit $\varepsilon \rightarrow 0^+$. We construct a nonlinear discontinous Galerkin scheme with space–time finite elements of general type that are discontinuous in time. Existence of a unique discrete solution is proven by application of Brouwer’s Theorem. We first derive abstract error estimates and then, for the case of piecewise polynomial finite elements, we prove an error in expectation of optimal order. All the appearing constants are estimated in terms of the parameter $\varepsilon $. Finally, we present a linear approximation of the nonlinear scheme, for which we prove existence of solution and optimal error in expectation in piecewise linear finite element spaces. The novelty of this work is based on the use of a finite element formulation in space and in time in $2+1$-dimensional subdomains for a nonlinear parabolic problem. In addition this problem involves noise. These types of schemes avoid any Runge–Kutta-type discretization for the evolutionary variable, and seem to be very effective when applied to equations of such a difficulty.

An algorithm for the simulation of curvilinear plane‐strain and axisymmetric hydraulic fractures with lag using the universal meshes

SummaryWe present an algorithm to simulate curvilinear hydraulic fractures in plane strain and axisymmetry. We restrict our attention to sharp fractures propagating in an isotropic, linear elastic medium and driven by the injection of a laminar, Newtonian fluid governed by lubrication theory, and we require the existence of a finite lag region between the fluid front and the crack tip. The key novelty of our approach is in how we discretize the evolving crack and fluid domains: we utilize universal meshes (UMs), a technique to create conforming triangulations of a problem domain by only perturbing nodes of a universal background mesh in the vicinity of the boundary. In this way, we construct meshes, which conform to the crack and to the fluid front. This allows us to build standard piecewise linear finite element spaces and to monolithically solve the quasistatic hydraulic fracture problem for the displacement field in the rock and the pressure in the fluid. We demonstrate the performance of our algorithms through three examples: a convergence study in plane strain, a comparison with experiments in axisymmetry, and a novel case of a fracture in a narrow pay zone.

An Error Analysis of Discontinuous Finite Element Methods for the Optimal Control Problems Governed by Stokes Equation

In this article, an abstract framework for the error analysis of discontinuous finite element method is developed for the distributed and Neumann boundary control problems governed by the stationary Stokes equation with control constraints. A priori error estimates of optimal order are derived for velocity and pressure in the energy norm and the L2-norm, respectively. Moreover, a reliable and efficient a posteriori error estimator is derived. The results are applicable to a variety of problems just under the minimal regularity possessed by the well-posedness of the problem. In particular, we consider the abstract results with suitable stable pairs of velocity and pressure spaces like as the lowest-order Crouzeix–Raviart finite element and piecewise constant spaces, piecewise linear and constant finite element spaces. The theoretical results are illustrated by the numerical experiments.

A Posteriori Error Analysis of the Crank–Nicolson Finite Element Method for Parabolic Integro-Differential Equations

We study a posteriori error analysis for the space-time discretizations of linear parabolic integro-differential equation in a bounded convex polygonal or polyhedral domain. The piecewise linear finite element spaces are used for the space discretization, whereas the time discretization is based on the Crank–Nicolson method. The Ritz–Volterra reconstruction operator (IMA J Numer Anal 35:341–371, 2015), a generalization of elliptic reconstruction operator (SIAM J Numer Anal 41:1585–1594, 2003), is used in a crucial way to obtain optimal rate of convergence in space. Moreover, a quadratic (in time) space-time reconstruction operator is introduced to establish second order convergence in time. The proposed method uses nested finite element spaces and the standard energy technique to obtain optimal order error estimator in the \(L^{\infty }(L^2)\)-norm. Numerical experiments are performed to validate the optimality of the error estimators.

A posteriori error analysis of the Crank–Nicolson finite element method for linear parabolic interface problems: A reconstruction approach

This article studies a residual-based a posteriori error analysis for the Crank–Nicolson time-stepping finite element method for a linear parabolic interface problem in a bounded convex polygonal domain in R 2 . A piecewise linear finite element space is used in space that is allowed to change in time and a modified Crank–Nicolson approximation is applied for the time discretizations. We employ a space–time reconstruction that is piecewise quadratic in time and the Clément-type interpolation estimates to derive optimal order in time and an almost optimal order in space a posteriori error bound in the L ∞ ( L 2 ) -norm. The interface is assumed to be of arbitrary shape but is of class C 2 for our purpose. Numerical results are presented to validate the derived estimators.

Adaptive techniques in SOLD methods

We present new results where free parameters in spurious oscillations at layers diminishing (SOLD) method are adaptively chosen. Provided numerical results are for conforming piecewise linear finite element space with free parameters from piecewise constant finite element space. We focus on a higher order convergence we discovered in previous paper by Lukáš (2015).

The backward euler anisotropic a posteriori error analysis for parabolic integro-differential equations

We derive two optimal a posteriori error estimators for an implicit fully discrete approximation to the solutions of linear integro-differential equations of the parabolic type. A continuous, piecewise linear finite element space is used for the space discretization and the time discretization is based on an implicit backward Euler method. The a posteriori error indicator corresponding to space discretization is derived using the anisotropic interpolation estimates in conjunction with a Zienkiewicz-Zhu error estimator to approach the error gradient. The error due to time discretization is derived using continuous, piecewise linear polynomial in time. We use the linear approximation of the Volterra integral term to estimate the quadrature error in the second estimator. Numerical experiments are performed on the isotropic mesh to validate the derived results.© 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1309–1330, 2016

Local Discontinuous Galerkin Methods Coupled with Implicit Integration Factor Methods for Solving Reaction-Cross-Diffusion Systems

We present a new numerical method for solving nonlinear reaction-diffusion systems with cross-diffusion which are often taken as mathematical models for many applications in the biological, physical, and chemical sciences. The two-dimensional system is discretized by the local discontinuous Galerkin (LDG) method on unstructured triangular meshes associated with the piecewise linear finite element spaces, which can derive not only numerical solutions but also approximations for fluxes at the same time comparing with most of study work up to now which has derived numerical solutions only. Considering the stability requirement for the explicit scheme with strict time step restriction (Δt=O(hmin2)), the implicit integration factor (IIF) method is employed for the temporal discretization so that the time step can be relaxed asΔt=O(hmin). Moreover, the method allows us to compute element by element and avoids solving a global system of nonlinear algebraic equations as the standard implicit schemes do, which can reduce the computational cost greatly. Numerical simulations about the system with exact solution and the Brusselator model, which is a theoretical model for a type of autocatalytic chemical reaction, are conducted to confirm the expected accuracy, efficiency, and advantages of the proposed schemes.

Convergence and Optimality of Higher-Order Adaptive Finite Element Methods for Eigenvalue Clusters

Proofs of convergence of adaptive finite element methods (AFEMs) for approximating eigenvalues and eigenfunctions of linear elliptic problems have been given in several recent papers. A key step in establishing such results for multiple and clustered eigenvalues was provided by Dai, He, and Zhou in [IMA J. Numer. Anal., 35 (2015), pp. 1934--1977], who proved convergence and optimality of AFEMs for eigenvalues of multiplicity greater than one. There it was shown that a theoretical (noncomputable) error estimator for which standard convergence proofs apply is equivalent to a standard computable estimator on sufficiently fine grids. In [Numer. Math., 130 (2015), pp. 467--496], Gallistl used a similar tool to prove that a standard AFEM for controlling eigenvalue clusters for the Laplacian using continuous piecewise linear finite element spaces converges with optimal rate. When considering either higher-order finite element spaces or nonconstant diffusion coefficients, however, the arguments of Dai, He, and Zhou and Gallistl do not yield equivalence of the practical and theoretical estimators for clustered eigenvalues. In this article we provide this missing key step, thus showing that standard adaptive FEMs for clustered eigenvalues employing elements of arbitrary polynomial degree converge with optimal rate. We additionally establish that a key user-defined input parameter in the AFEM, the bulk marking parameter, may be chosen entirely independently of the properties of the target eigenvalue cluster. All of these results assume a fineness condition on the initial mesh in order to ensure that the nonlinearity is sufficiently resolved.

A nonlinear weighted least-squares finite element method for the Carreau–Yasuda non-Newtonian model

We study a nonlinear weighted least-squares finite element method for the Navier–Stokes equations governing non-Newtonian fluid flows by using the Carreau–Yasuda model. The Carreau–Yasuda model is used to describe the shear-thinning behavior of blood. We prove that the least-squares approximation converges to linearized solutions of the non-Newtonian model at the optimal rate. By using continuous piecewise linear finite element spaces for all variables and by appropriately adjusting the nonlinear weighting function, we obtain optimal L2-norm error convergence rates in all variables. Numerical results are given for a Carreau fluid in the 4-to-1 contraction problem, revealing the shear-thinning behavior. The physical parameter effects are also investigated.

An auxiliary space multigrid preconditioner for the weak Galerkin method

In this paper, we construct an auxiliary space multigrid preconditioner for the weak Galerkin method for second-order diffusion equations, discretized on simplicial 2D or 3D meshes. The idea of the auxiliary space multigrid preconditioner is to use an auxiliary space as a “coarse” space in the multigrid algorithm, where the discrete problem in the auxiliary space can be easily solved by an existing solver. In our construction, we conveniently use the H1 conforming piecewise linear finite element space as an auxiliary space. The main technical difficulty is to build the connection between the weak Galerkin discrete space and the H1 conforming piecewise linear finite element space. We successfully constructed such an auxiliary space multigrid preconditioner for the weak Galerkin method, as well as the reduced system of the weak Galerkin method involving only the degrees of freedom on edges/faces. The preconditioned systems are proved to have condition numbers independent of the mesh size. Numerical experiments further support the theoretical results.

A subspace correction method for discontinuous Galerkin discretizations of linear elasticity equations

We study preconditioning techniques for discontinuous Galerkin discretizations of isotropic linear elasticity problems in primal (displacement) formulation. We propose subspace correction methods based on a splitting of the vector valued piecewise linear discontinuous finite element space, that are optimal with respect to the mesh size and the Lame parameters. The pure displacement, the mixed and the traction free problems are discussed in detail. We present a convergence analysis of the proposed preconditioners and include numerical examples that validate the theory and assess the performance of the preconditioners.