Piecewise Polynomials Of Degree Research Articles

Hybridizable Discontinuous Galerkin Methods for Timoshenko Beams

In this paper, we introduce a new class of discontinuous Galerkin methods for Timoshenko beams. The main feature of these methods is that they can be implemented in an efficient way through a hybridization procedure which reduces the globally coupled unknowns to approximations to the displacement and bending moment at the element boundaries. After displaying the methods, we obtain conditions under which they are well defined. We then compare these new methods with the already existing discontinuous Galerkin methods for Timoshenko beams. Finally, we display extensive numerical results to ascertain the influence of the stabilization parameters on the accuracy of the approximation. In particular, we find specific choices for which all the variables, namely, the displacement, the rotation, the bending moment and the shear force converge with the optimal order of k+1 when each of their approximations are taken to be piecewise polynomial of degree k?0.

Mixed Finite Element Methods for Incompressible Flow: Stationary Navier–Stokes Equations

In [Z. Cai, C. Tong, P. S. Vassilevski, and C. Wang, Numer. Methods Partial Differential Equations, to appear], the authors developed and analyzed a mixed finite element method for the stationary Stokes equations based on the pseudostress-velocity formulation. The pseudostress and the velocity are approximated by a stable pair of finite elements: Raviart–Thomas elements of index $k\geq0$ and discontinuous piecewise polynomials of degree $k\geq0$, respectively. This paper extends the method to the stationary, incompressible Navier–Stokes equations. Under appropriate assumptions, we show that the pseudostress-velocity formulation of the Navier–Stokes equation and its discrete counterpart have branches of nonsingular solutions, and error estimates of the mixed finite element approximations are established as well.

Extrapolation Using a Modified Projection Method

In [14], a new method based on projections onto a space of piecewise polynomials of degree ≤r − 1 has been shown to give a convergence of order 4r for second-kind integral equations. The size of the system of equations that must be solved, in implementing this method, remains the same as for the Galerkin/collocation method. In this article the solution obtained by the proposed method is shown to have an asymptotic series expansion which remains valid in the discrete version. The Richardson extrapolation can then be used to further improve the order of convergence to 4r + 2.

Analysis of a velocity–pressure–pseudostress formulation for the stationary Stokes equations

We consider a non-standard mixed approach for the Stokes problem in which the velocity, the pressure, and the pseudostress are the main unknowns. Alternatively, the pressure can be eliminated from the original equations, thus yielding an equivalent formulation with only two unknowns. In this paper we develop a priori and a posteriori error analyses of both approaches. We first apply the Babuška–Brezzi theory to prove the well-posedness of the continuous and discrete formulations. In particular, we show that Raviart–Thomas elements of order k ⩾ 0 for the pseudostresses, and piecewise polynomials of degree k for the velocities and the pressures, ensure unique solvability and stability of the associated Galerkin schemes. Then, we derive reliable and efficient residual-based a posteriori error estimators for both schemes, without and with pressure unknown. Finally, we provide several numerical results illustrating the good performance of the resulting mixed finite element methods, confirming the theoretical properties of the estimators, and showing the behaviour of the associated adaptive algorithms.

Symmetric multi-box splines for higher degree

We consider the space Sn=Sn(v0,…,vn+r) of compactly supported Cn−1 piecewise polynomials on a mesh M of lines through ℤ2 in directions v0,…,vn+r. A sequence ψ=(ψ1,…,ψr) of elements of Sn is called a multi-box spline if every element of Sn is a finite linear combination of shifts of (the components of) ψ. For the case n=2, 3 we give some examples for multi-box splines and show that they are not always stable. It is further shown that any Cn−1 piecewise polynomial of degree n≥2 on M, is possibly a symmetric multi-box spline.

Explicit polynomial preserving trace liftings on a triangle

AbstractWe give an explicit formula for a right inverse of the trace operator from the Sobolev space H1(T) on a triangle T to the trace space H1/2(∂T) on the boundary. The lifting preserves polynomials in the sense that if the boundary data are piecewise polynomial of degree N, then the lifting is a polynomial of total degree at most N and the lifting is shown to be uniformly stable independently of the polynomial order. Moreover, the same operator is shown to provide a uniformly stable lifting from L2(∂T) to H1/2(T). Finally, the lifting is used to construct a uniformly bounded right inverse for the normal trace operator from the space H(div; T) to H–1/2(∂T) which also preserves polynomials. Applications to the analysis of high order numerical methods for partial differential equations are indicated (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Discontinuous Galerkin method for an evolution equation with a memory term of positive type

We consider an initial value problem for a class of evolution equations incorporating a memory term with a weakly singular kernel bounded by C ( t − s ) α − 1 C(t-s)^{\alpha -1} , where 0 > α > 1 0>\alpha >1 . For the time discretization we apply the discontinuous Galerkin method using piecewise polynomials of degree at most q − 1 q-1 , for q = 1 q=1 or 2 2 . For the space discretization we use continuous piecewise-linear finite elements. The discrete solution satisfies an error bound of order k q + h 2 ℓ ( k ) k^q+h^2\ell (k) , where k k and h h are the mesh sizes in time and space, respectively, and ℓ ( k ) = max ( 1 , log ⁡ k − 1 ) \ell (k)=\max (1,\log k^{-1}) . In the case q = 2 q=2 , we prove a higher convergence rate of order k 3 + h 2 ℓ ( k ) k^3+h^2\ell (k) at the nodes of the time mesh. Typically, the partial derivatives of the exact solution are singular at t = 0 t=0 , necessitating the use of non-uniform time steps. We compare our theoretical error bounds with the results of numerical computations.

On the relationship of local projection stabilization to other stabilized methods for one-dimensional advection–diffusion equations

We consider the one-level approach of the local projection stabilization (LPS) for solving a singularly perturbed advection–diffusion two-point boundary value problem. Eliminating the enrichments we end up with the differentiated residual method (DRM) which coincides for piecewise linears with the streamline upwind Petrov–Galerkin (SUPG) method and for piecewise polynomials of degree r ⩾ 2 with the variational multiscale method (VMS). Furthermore, we show that in certain cases the stabilization parameter can be chosen in such a way that the piecewise linear part of the solution becomes nodal exact. In this way, we obtain explicit formulas for the stabilization parameter depending on the local meshsize, the polynomial degree r of the approximation space, and the data of the problem. Finally, we discuss the behaviour of different modes of the discrete solution when varying the stabilization parameter.

Sparse finite element approximation of high-dimensional transport-dominated diffusion problems

We develop the analysis of stabilized sparse tensor-product finite element methods for high-dimensional, non-self-adjoint and possibly degenerate second-order partial differential equations of the form , , where is a symmetric positive semidefinite matrix, using piecewise polynomials of degree p ≥ 1. Our convergence analysis is based on new high-dimensional approximation results in sparse tensor-product spaces. We show that the error between the analytical solution u and its stabilized sparse finite element approximation uh on a partition of Ω of mesh size h = hL = 2-L satisfies the following bound in the streamline-diffusion norm , provided u belongs to the space of functions with square-integrable mixed (k+1)st derivatives: where , i=0,1, and . We show, under various mild conditions relating L to p, L to d, or p to d, that in the case of elliptic transport-dominated diffusion problems , and hence for p ≥ 1 the 'error constant' exhibits exponential decay as d → ∞; in the case of a general symmetric positive semidefinite matrix a, the error constant is shown to grow no faster than . In any case, in the absence of assumptions that relate L, p and d, the error is still bounded by , where for all L, p, d ≥ 2.

A unified framework for the construction of one-step finite volume and discontinuous Galerkin schemes on unstructured meshes

In this article, a conservative least-squares polynomial reconstruction operator is applied to the discontinuous Galerkin method. In a first instance, piecewise polynomials of degree N are used as test functions as well as to represent the data in each element at the beginning of a time step. The time evolution of these data and the flux computation, however, are then done with a different set of piecewise polynomials of degree M ⩾ N , which are reconstructed from the underlying polynomials of degree N. This approach yields a general, unified framework that contains as two special cases classical high order finite volume (FV) schemes ( N = 0 ) as well as the usual discontinuous Galerkin (DG) method ( N = M ) . In the first case, the polynomial is reconstructed from cell averages, for the latter, the reconstruction reduces to the identity operator. A completely new class of numerical schemes is generated by choosing N ≠ 0 and M > N . The reconstruction operator is implemented for arbitrary polynomial degrees N and M on unstructured triangular and tetrahedral meshes in two and three space dimensions. To provide a high order accurate one-step time integration of the same formal order of accuracy as the spatial discretization operator, the (reconstructed) polynomial data of degree M are evolved in time locally inside each element using a new local continuous space–time Galerkin method. As a result of this approach, we obtain, as a high order accurate predictor, space–time polynomials for the vector of conserved variables and for the physical fluxes and source terms, which then can be used in a natural way to construct very efficient fully-discrete and quadrature-free one-step schemes. This feature is particularly important for DG schemes in three space dimensions, where the cost of numerical quadrature may become prohibitively expensive for very high orders of accuracy. Numerical convergence studies of all members of the new general class of proposed schemes are shown up to sixth-order of accuracy in space and time on unstructured two- and three-dimensional meshes for two very prominent nonlinear hyperbolic systems, namely for the Euler equations of compressible gas dynamics and the equations of ideal magnetohydrodynamics (MHD). The results indicate that the new class of intermediate schemes ( N ≠ 0 , M > N ) is computationally more efficient than classical finite volume or DG schemes. Finally, a large set of interesting test cases is solved on unstructured meshes, where the proposed new time stepping approach is applied to the equations of ideal and relativistic MHD as well as to nonlinear elasticity, using a standard high order WENO finite volume discretization in space to cope with discontinuous solutions.

Mixed Discontinuous Galerkin Finite Element Method for the Biharmonic Equation

In this paper, we first split the biharmonic equation Δ2 u=f with nonhomogeneous essential boundary conditions into a system of two second order equations by introducing an auxiliary variable v=Δu and then apply an hp-mixed discontinuous Galerkin method to the resulting system. The unknown approximation v h of v can easily be eliminated to reduce the discrete problem to a Schur complement system in u h , which is an approximation of u. A direct approximation v h of v can be obtained from the approximation u h of u. Using piecewise polynomials of degree p?3, a priori error estimates of u?u h in the broken H 1 norm as well as in L 2 norm which are optimal in h and suboptimal in p are derived. Moreover, a priori error bound for v?v h in L 2 norm which is suboptimal in h and p is also discussed. When p=2, the preset method also converges, but with suboptimal convergence rate. Finally, numerical experiments are presented to illustrate the theoretical results.

A family of 3D continuously differentiable finite elements on tetrahedral grids

A family of continuously differentiable piecewise polynomials of degree 9 and higher, on general tetrahedral grids, is constructed, by simplifying and extending the P 9 element of Ženišek. A mathematical justification and numerical tests are presented. The current computing power is still limited for the computation with 3D C 1 finite elements in general. The construction here mainly serves the purposes of understanding and ensuring the approximation properties of C 1 finite elements spaces on tetrahedral grids. In particular, this construction indicates that the 3D divergence-free C 0 - P k elements have the full order of approximation for any degree k ⩾ 8 .

Hp-Discontinuous Galerkin methods for strongly nonlinear elliptic boundary value problems

In this paper, we have analyzed a one parameter family of hp-discontinuous Galerkin methods for strongly nonlinear elliptic boundary value problems $$-\nabla \cdot {\rm a} (u, \nabla u) + f (u, \nabla u) = 0$$ with Dirichlet boundary conditions. These methods depend on the values of the parameter $$\theta\in[-1,1]$$, where θ = + 1 corresponds to the nonsymmetric and θ = −1 corresponds to the symmetric interior penalty methods when $${\rm a}(u,\nabla u)={\nabla}u$$ and f(u,∇u) = −f, that is, for the Poisson problem. The error estimate in the broken H 1 norm, which is optimal in h (mesh size) and suboptimal in p (degree of approximation) is derived using piecewise polynomials of degree p ≥ 2, when the solution $$u\in H^{5/2}(\Omega)$$. In the case of linear elliptic problems also, this estimate is optimal in h and suboptimal in p. Further, optimal error estimate in the L 2 norm when θ = −1 is derived. Numerical experiments are presented to illustrate the theoretical results.

Improving the Rate of Convergence of High-Order Finite Elements on Polyhedra II: Mesh Refinements and Interpolation

We construct a sequence of meshes 𝒯 k ′ that provides quasi-optimal rates of convergence for the solution of the Poisson equation on a bounded polyhedral domain with right-hand side in H m−1, m ≥ 2. More precisely, let Ω ⊂ ℝ3 be a bounded polyhedral domain and let u ∈ H 1(Ω) be the solution of the Poisson problem − Δ u = f ∈ H m−1(Ω), m ≥ 2, u = 0 on ∂ Ω. Also, let S k be the finite element space of continuous, piecewise polynomials of degree m ≥ 2 on 𝒯 k ′ and let u k ∈ S k be the finite element approximation of u, then ‖u − u k ‖ H 1(Ω) ≤ C dim(S k )−m/3 ‖f‖ H m−1(Ω), with C independent of k and f. Our method relies on the a priori estimate ‖u‖𝒟 ≤ C ‖f‖ H m−1(Ω) in certain anisotropic weighted Sobolev spaces , with a > 0 small, determined only by Ω. The weight is the distance to the set of singular boundary points (i.e., edges). The main feature of our mesh refinement is that a segment AB in 𝒯 k ′ will be divided into two segments AC and CB in 𝒯 k+1′ as follows: |AC| = |CB| if A and B are equally singular and |AC| = κ |AB| if A is more singular than B. We can choose κ ≤ 2−m/a . This allows us to use a uniform refinement of the tetrahedra that are away from the edges to construct 𝒯 k ′.

Locking-free Reissner–Mindlin elements without reduced integration

In a recent paper of Arnold et al. [D.N. Arnold, F. Brezzi, L.D. Marini, A family of discontinuous Galerkin finite elements for the Reissner–Mindlin plate, J. Sci. Comput. 22 (2005) 25–45], the ideas of discontinuous Galerkin methods were used to obtain and analyze two new families of locking free finite element methods for the approximation of the Reissner–Mindlin plate problem. By following their basic approach, but making different choices of finite element spaces, we develop and analyze other families of locking free finite elements that eliminate the need for the introduction of a reduction operator, which has been a central feature of many locking-free methods. For k ⩾ 2 , all the methods use piecewise polynomials of degree k to approximate the transverse displacement and (possibly subsets) of piecewise polynomials of degree k − 1 to approximate both the rotation and shear stress vectors. The approximation spaces for the rotation and the shear stress are always identical. The methods vary in the amount of interelement continuity required. In terms of smallest number of degrees of freedom, the simplest method approximates the transverse displacement with continuous, piecewise quadratics and both the rotation and shear stress with rotated linear Brezzi–Douglas–Marini elements.

Optimal Superconvergence Results for Delay Integro‐Differential Equations of Pantograph Type

We analyze the optimal (global and local) orders of superconvergence of collocation solutions $u_{h}$ on uniform meshes $I_{h}$ for delay Volterra integro-differential equations with proportional delay functions given by $\theta (t) = qt (0 < q < 1, t \in [0,T])$. In particular, we show that if $u_{h}$ is a continuous piecewise polynomial of degree $m \geq 2$, and if collocation is at the Gauss (-Legendre) points, then the (optimal) order of local superconvergence on $I_{h}$ is $p^{\ast } = m+2$. It turns out that the same order $p^{\ast }$ holds for nonlinear (strictly increasing) delay functions vanishing at $t = 0$. However, on judiciously chosen geometric meshes, collocation at the Gauss points yields the order $2m-\varepsilon _{N}$, where $\varepsilon _{N} \rightarrow 0$ as the number $N$ of mesh points tends to infinity. Optimal local superconvergence results for the pantograph delay differential equation are obtained as special cases of our general analysis.

Superconvergent Derivative Recovery for Lagrange Triangular Elements of Degreepon Unstructured Grids

In this paper, we develop a postprocessing derivative recovery scheme for the finite element solution $u_h$ on general unstructured but shape regular triangulations. In the case of continuous piecewise polynomials of degree $p\geq 1$, by applying the global $L^2$ projection ($Q_h$) and a smoothing operator ($S_h$), the recovered $p$th derivatives ($S_h^m Q_h\partial^p u_h$) superconverge to the exact derivatives ($\partial^p u$). Based on this technique we are able to derive a local error indicator depending only on the geometry of corresponding element and the $(p+1)$st derivatives approximated by $\partial S_h^m Q_h\partial^p u_h$. We provide several numerical examples illustrating the effectiveness of our schemes. We also observe that higher order elements are likely to require more conservative refinement strategies to create meshes corresponding to optimal orders of convergence.

Univariate splines: Equivalence of moduli of smoothness and applications

Several results on equivalence of moduli of smoothness of univariate splines are obtained. For example, it is shown that, for any 1 < k ≤ r +1, 0 ≤ m ≤ r-1, and 1 ≤ p < ∞, the inequality n -v ω k-v (s (v) , n -1 ) p ∼ ω k (s,n -1 ) p , 1 ≤ v ≤ min{k, m + 1}, is satisfied, where s ∈ C m [-1, 1] is a piecewise polynomial of degree ≤ r on a quasi-uniform (i.e., the ratio of lengths of the largest and the smallest intervals is bounded by a constant) partition of an interval. Similar results for Chebyshev partitions and weighted Ditzian-Totik moduli of smoothness are also obtained. These results yield simple new constructions and allow considerable simplification of various known proofs in the area of constrained approximation by polynomials and splines.

A new superconvergent collocation method for eigenvalue problems

Here we propose a new method based on projections for the approximate solution of eigenvalue problems. For an integral operator with a smooth kernel, using an interpolatory projection at Gauss points onto the space of (discontinuous) piecewise polynomials of degree ≤ r − 1 \leq r-1 , we show that the proposed method exhibits an error of the order of 4 r 4r for eigenvalue approximation and of the order of 3 r 3r for spectral subspace approximation. In the case of a simple eigenvalue, we show that by using an iteration technique, an eigenvector approximation of the order 4 r 4r can be obtained. This improves upon the order 2 r 2r for eigenvalue approximation in the collocation/iterated collocation method and the orders r r and 2 r 2r for spectral subspace approximation in the collocation method and the iterated collocation method, respectively. We illustrate this improvement in the order of convergence by numerical examples.

Discontinuous Galerkin finite element approximation of quasilinear elliptic boundary value problems I: the scalar case

We develop a one-parameter family of hp-version discontinuous Galerkin finite element methods, parameterised by θ ∈ [−1, 1], for the numerical solution of quasilinear elliptic equations in divergence form on a bounded open set Ω ⊂ ℝd, d ≥ 2. In particular, we consider the analysis of the family for the equation −∇ ·{μ(x, |∇u|)∇u} = f(x) subject to mixed Dirichlet–Neumann boundary conditions on ∂ Ω. It is assumed that μ is a real-valued function, μ ∈ C(Ω̄ × [0, ∞)), and there exist positive constants mμ and Mμ such that mμ(t − s) ≤ μ(x, t)t − μ(x, s)s ≤ Mμ(t − s) for t ≥ s ≥ 0 and all x ∈ Ω̄. Using a result from the theory of monotone operators for any value of θ ∈ [−1, 1], the corresponding method is shown to have a unique solution uDG in the finite element space. If u ∈ C1(Ω) ∩ Hk(Ω), k ≥ 2, then with discontinuous piecewise polynomials of degree p ≥ 1, the error between u and uDG, measured in the broken H1(Ω)-norm, is 𝒪(hs−1/pk−3/2), where 1 ≤ s ≤ min {p + 1, k}.