On the finite volume element method

Summary The finite volume element method (FVE) is a discretization technique for partial differential equations. It uses a volume integral formulation of the problem with a finite partitioning set of volumes to discretize the equations, then restricts the admissible functions to a finite element space to discretize the solution. this paper develops discretization error estimates for general selfadjoint elliptic boundary value problems with FVE based on triangulations with linear finite element spaces and a general type of control volume. We establishO(h) estimates of the error in a discreteH ¹ semi-norm. Under an additional assumption of local uniformity of the triangulation the estimate is improved toO(h ²). Results on the effects of numerical integration are also included.This research was sponsored in part by the Air Force Office of Scientific Research under grant number AFOSR-86-0126 and the National Science Foundation under grant number DMS-8704169. This work was performed while the author was at the University of Colorado at Denver     

Stable enrichment and local preconditioning in the particle-partition of unity method

This paper is concerned with the stability and approximation properties of enriched meshfree and generalized finite element methods. In particular we focus on the particle-partition of unity method (PPUM) yet the presented results hold for any partition of unity based enrichment scheme. The goal of our enrichment scheme is to recover the optimal convergence rate of the uniform h-version independent of the regularity of the solution. Hence, we employ enrichment not only for modeling purposes but rather to improve the approximation properties of the numerical scheme. To this end we enrich our PPUM function space in an enrichment zone hierarchically near the singularities of the solution. This initial enrichment however can lead to a severe ill-conditioning and can compromise the stability of the discretization. To overcome the ill-conditioning of the enriched shape functions we present an appropriate local preconditioner which yields a stable and well-conditioned basis independent of the employed initial enrichment. The construction of this preconditioner is of linear complexity with respect to the number of discretization points. We obtain optimal error bounds for an enriched PPUM discretization with local preconditioning that are independent of the regularity of the solution globally and within the employed enrichment zone we observe a kind of super-convergence. The results of our numerical experiments clearly show that our enriched PPUM with local preconditioning recovers the optimal convergence rate of O(h ^p ) of the uniform h-version globally. For the considered model problems from linear elastic fracture mechanics we obtain an improved convergence rate of O(h ^p+δ ) with d 3 \frac12{\delta\geq\frac{1}{2}} for p = 1. The convergence rate of our multilevel solver is essentially the same for a purely polynomial approximation and an enriched approximation.     

Error Estimates for Finite Element Approximations of a Viscous Wave Equation

We consider a family of fully discrete finite element schemes for solving a viscous wave equation, where the time integration is based on the Newmark method. A rigorous stability analysis based on the energy method is developed. Optimal error estimates in both time and space are obtained. For sufficiently smooth solutions, it is demonstrated that the maximal error in the L ²-norm over a finite time interval converges optimally as O(h ^p+1 + Δt ^s ), where p denotes the polynomial degree, s = 1 or 2, h the mesh size, and Δt the time step.     

On solenoidal high‐degree polynomial approximations to solutions of the stationary Stokes equations

The cell discretization algorithm, a nonconforming extension of the finite element method, is used to obtain approximations to the velocity and pressure functions satisfying the Stokes equations. Error estimates show convergence of the method. An implementation using polynomial bases is described that permits the use of the continuous approximations of the h‐p finite element method and exactly satisfies the solenoidal requirement. We express the error estimates in terms of the diameter h of a cell and degree p of the approximation on each cell. Examples of 10^th degree polynomial approximations are described that substantiate the theoretical estimates. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 480–493, 2000     

On approximating the solution of the non-stationary Stokes equations using the cell discretization algorithm

The cell discretization algorithm, a nonconforming extension of the finite element method, is used to obtain approximations to the velocity and pressure satisfying the nonstationary Stokes equations. Error estimates show convergence of the approximations. An implementation using polynomial bases is described that permits the use of the continuous approximations of the h–p finite element method and exactly satisfies the solenoidal requirement. We express the error estimates in terms of the diameter h of a cell and the degree p of the approximation on each cell. Results of an experiment with p10 are presented that confirm the theoretical estimates.     

Error bounds for multistep methods revisited

For linear multistep methods with constant stepsize we consider error bounds in terms of weightedL ₂-norms ofh ^px^(p) rather than ofh ^px^(p+1). The bounds apply to stiff systemsx'=Ax+f(t,x) where the spectrum ofA lies in a sector andf is of moderate size.     

Stability and convergence of fully discrete finite element schemes for the acoustic wave equation

In this paper, we investigate the stability and convergence of some fully discrete finite element schemes for solving the acoustic wave equation where a discontinuous Galerkin discretization in space is used. We first review and compare conventional time-stepping methods for solving the acoustic wave equation. We identify their main properties and investigate their relationship. The study includes the Newmark algorithm which has been used extensively in applications. We present a rigorous stability analysis based on the energy method and derive sharp stability results covering some well-known CFL conditions. A convergence analysis is carried out and optimal a priori error estimates are obtained. For sufficiently smooth solutions, we demonstrate that the maximal error in the L ²-norm error over a finite time interval converges optimally as O(h ^p+1+??t ^s ), where p denotes the polynomial degree, s=1 or 2, h the mesh size, and ??t the time step.     

Error estimates using the cell discretization method for second-order hyperbolic equations

The cell discretization algorithm provides approximate solutions to second-order hyperbolic equations with coefficients independent of time. We obtain error estimates that show general convergence for homogeneous problems using semi-discrete approximations. A polynomial implementation of the algorithm is described that is a nonconforming extension of the finite element method that can also produce the continuous approximations of an h–p finite element method. Numerical tests are made that confirm the theoretical estimates. © 1997 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13: 531–548, 1997     

Finite element method for Grwünwald–Letnikov time-fractional partial differential equation

In this article, we consider the finite element methods (FEM) for Grwünwald–Letnikov time-fractional diffusion equation, which is obtained from the standard two-dimensional diffusion equation by replacing the first-order time derivative with a fractional derivative (of order α, with 0?h ^r+1?+?τ^2-α), where h, τ and r are the space step size, time step size and polynomial degree, respectively. A numerical example is presented to verify the order of convergence.     

Analysis of the immersed boundary method for a finite element Stokes problem

Convergence results are presented for the immersed boundary (IB) method applied to a model Stokes problem. As a discretization method, we use the finite element method. First, the immersed force field is approximated using a regularized delta function. Its error in the W^{?1, p} norm is examined for 1 ≤ p < n/(n ? 1), with n representing the space dimension. Subsequently, we consider IB discretization of the Stokes problem and examine the regularization and discretization errors separately. Consequently, error estimate of order h^{1 ? α} in the W^{1, 1} × L¹ norm for the velocity and pressure is derived, where α is an arbitrary small positive number. The validity of those theoretical results is confirmed from numerical examples.     

A Priori Error Estimates for Finite Element Methods for H (2,1)-Elliptic Equations

The convergence of finite element methods for linear elliptic boundary value problems of second and forth order is well understood. In this article, we introduce finite element approximations of some linear semi-elliptic boundary value problem of mixed order on a two-dimensional rectangular domain Q. The equation is of second order in one direction and forth order in the other and appears in the optimal control of parabolic partial differential equations if one eliminates the control and the state (or the adjoint state) in the first order optimality conditions. We establish a regularity result and estimate for the finite element error of conforming approximations of this equation. The finite elements in use have a tensor product structure, in one dimension we use linear, quadratic or cubic Lagrange elements in the other dimension cubic Hermite elements. For these elements, we prove the error bound O(h ² + τ ^k ) in the energy norm and O((h ² + τ ^k )(h ² + τ)) in the L ²(Q)-norm.     

Galerkin discontinuous approximation of the transport equation and viscoelastic fluid flow on quadrilaterals

Numerical simulation of industrial processes involving viscoelastic liquids is often based on finite element methods on quadrilateral meshes. However, numerical analysis of these methods has so far been limited to triangular meshes. In this work, we consider quadrilateral meshes. We first study the approximation of the transport equation by a Galerkin discontinuous method and prove an 𝒪(h^k+1/2) error estimates for the Q_k finite element. Then we study a differential model for viscoelastic flow with unknowns u the velocity, p the pressure, and σ the viscoelastic part of the extra-stress tensor. The approximations are ((Q₁)² transforms of) Q_k+1 continuous for u, Q_k discontinuous for σ, and P_k discontinuous for p, with k ≥ 1. Upwinding for σ is obtained by the Galerkin discontinuous method. We show that an error estimate of order 𝒪(h^k+1/2) is valid in the energy norm for the three unknowns. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 97–114, 1998     

A posteriori error estimation with finite element methods of lines for one-dimensional parabolic systems

Summary Consider the solution of one-dimensional linear initial-boundary value problems by a finite element method of lines using a piecewiseP ^th -degree polynomial basis. A posteriori estimates of the discretization error are obtained as the solutions of either local parabolic or local elliptic finite element problems using piecewise polynomial corrections of degreep+1 that vanish at element ends. Error estimates computed in this manner are shown to converge in energy under mesh refinement to the exact finite element discretization error. Computational results indicate that the error estimates are robust over a wide range of mesh spacings and polynomial degrees and are, furthermore, applicable in situations that are not supported by the analysis.This research was partially supported by the U.S. Air Force Office of Scientific Research, Air Force Systems Command, USAF, under Grant Number AFOSR 90-0194; by the U.S. Army Research Office under Contract Number DAAL03-91-G-0215; and by the National Science Foundation under Institutional Infrastructure Grant Number CDA-8805910     

What is the smallest possible constant in Céa’s lemma?

We consider finite element approximations of a second order elliptic problem on a bounded polytopic domain in ℝ^d with d ∈ {1, 2, 3, ...} The constant C ⩾ 1 appearing in Céa’s lemma and coming from its standard proof can be very large when the coefficients of an elliptic operator attain considerably different values. We restrict ourselves to regular families of uniform partitions and linear simplicial elements. Using a lower bound of the interpolation error and the supercloseness between the finite element solution and the Lagrange interpolant of the exact solution, we show that the ratio between discretization and interpolation errors is equal to as the discretization parameter h tends to zero. Numerical results in one and two-dimensional case illustrating this phenomenon are presented. This research was supported by Shandong Province Young Scientists Foundation of China 2005BS01008, Institutional Research Plan AV02 101 90503, and by Grant No A 1019201 of the Academy of Sciences of the Czech Republic.     

A second order explicit finite element scheme to multidimensional conservation laws and its convergence

A second order explicit finite element scheme is given for the numerical computation to multi-dimensional scalar conservation laws.L ^p convergence to entropy solutions is proved under some usual conditions. For two-dimensional problems, uniform mesh, and sufficiently smooth solutions a second order error estimate inL ² is proved under a stronger condition, Δt≤Ch ^2/4     

A second-order accurate numerical method for a fractional wave equation

We study a generalized Crank–Nicolson scheme for the time discretization of a fractional wave equation, in combination with a space discretization by linear finite elements. The scheme uses a non-uniform grid in time to compensate for the singular behaviour of the exact solution at t = 0. With appropriate assumptions on the data and assuming that the spatial domain is convex or smooth, we show that the error is of order k ² + h ², where k and h are the parameters for the time and space meshes, respectively.     

A priori error estimates of an extrapolated space‐time discontinuous galerkin method for nonlinear convection‐diffusion problems

We deal with the numerical solution of a scalar nonstationary nonlinear convection‐diffusion equation. We employ a combination of the discontinuous Galerkin finite element (DGFE) method for the space as well as time discretization. The linear diffusive and penalty terms are treated implicitly whereas the nonlinear convective term is treated by a special higher order explicit extrapolation from the previous time step, which leads to the necessity to solve only a linear algebraic problem at each time step. We analyse this scheme and derive a priori asymptotic error estimates in the L^∞(L²) –norm and the L²(H¹) –seminorm with respect to the mesh size h and time step τ. Finally, we present an efficient solution strategy and numerical examples verifying the theoretical results. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1456–1482, 2010     

Boundary concentrated finite elements for optimal boundary control problems of elliptic PDEs

We investigate the discretization of optimal boundary control problems for elliptic equations on two-dimensional polygonal domains by the boundary concentrated finite element method. We prove that the discretization error ||u^*-u_h^*||_L²(G)\|u^{*}-u_{h}^{*}\|_{L^{2}(\Gamma)} decreases like N ⁻¹, where N is the total number of unknowns. This makes the proposed method favorable in comparison to the h-version of the finite element method, where the discretization error behaves like N ^−3/4 for uniform meshes. Moreover, we present an algorithm that solves the discretized problem in almost optimal complexity. The paper is complemented with numerical results.     

Some superconvergence results of high-degree finite element method for a second order elliptic equation with variable coefficients

In this paper, some superconvergence results of high-degree finite element method are obtained for solving a second order elliptic equation with variable coefficients on the inner locally symmetric mesh with respect to a point x ₀ for triangular meshes. By using of the weak estimates and local symmetric technique, we obtain improved discretization errors of O(h ^p+1 |ln h|²) and O(h ^p+2 |ln h|²) when p (≥ 3) is odd and p (≥ 4) is even, respectively. Meanwhile, the results show that the combination of the weak estimates and local symmetric technique is also effective for superconvergence analysis of the second order elliptic equation with variable coefficients.     

Hyperbolic conservation laws on manifolds. An error estimate for finite volume schemes

Following Ben-Artzi and LeFloch, we consider nonlinear hyperbolic conservation laws posed on a Riemannian manifold, and we establish an L1-error estimate for a class of finite volume schemes allowing for the approximation of entropy solutions to the initial value problem. The error in the L1 norm is of order h1/4 at most, where h represents the maximal diameter of elements in the family of geodesic triangulations. The proof relies on a suitable generalization of Cockburn, Coquel, and LeFloch＇s theory which was originally developed in the Euclidian setting. We extend the arguments to curved manifolds, by taking into account the effects to the geometry and overcoming several new technical difficulties.