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We show that the discrete velocity solution converges at the optimal order when solving the steady state Stokes equations by the Pk$$ {P}_k $$‐Pk−1disc$$ {P}_{k-1}^{\mathrm{disc}} $$ mixed finite element method for k≥4$$ k\ge 4 $$ on 2D triangular grids or k≥6$$ k\ge 6 $$ on tetrahedral grids, even in the case the inf‐sup condition fails. By a simple L2$$ {L}^2 $$‐projection of the discrete Pk−1$$ {P}_{k-1} $$ pressure to the space of continuous Pk−1$$ {P}_{k-1} $$ polynomials, we show this post‐processed pressure solution also converges at the optimal order. Both 2D and 3D numerical tests are presented, verifying the theory. [ABSTRACT FROM AUTHOR]

A conforming discontinuous Galerkin (CDG) finite element is constructed for solving second order elliptic equations on polygonal and polyhedral meshes. The numerical trace on the edge between two elements is no longer the average of two discontinuous Pk functions on the two sides, but a lifted Pk+2$P_{k+2}$ function from four Pk functions. When the numerical gradient space is the H(div,T)$H(\hbox{div},T)$ subspace of piecewise Pk+1d$P_{k+1}^d$ polynomials on subtriangles/subtehrahedra of a polygon/polyhedron T$T$ which have a one‐piece polynomial divergence on T$T$, this CDG method has a superconvergence of order two above the optimal order. Due to the superconvergence, we define a post‐process which lifts a Pk CDG solution to a quasi‐optimal Pk+2$P_{k+2}$ solution on each element. Numerical examples in 2D and 3D are computed and the results confirm the theory. [ABSTRACT FROM AUTHOR]

The Neilan Pk$$ {P}_k $$‐Pk−1$$ {P}_{k-1} $$ divergence‐free finite element is stable on any tetrahedral grid, where the piece‐wise Pk$$ {P}_k $$ polynomial velocity is C0$$ {C}^0 $$ on the grid, C1$$ {C}^1 $$ on edges and C2$$ {C}^2 $$ at vertices, and the piece‐wise Pk−1$$ {P}_{k-1} $$ polynomial pressure is C0$$ {C}^0 $$ on edges and C1$$ {C}^1 $$ at vertices. However the method does not work if the exact pressure solution does not vanish on all domain edges, because of the excessive continuity requirements. We extend the Neilan element by removing the extra requirements at domain boundary edges. That is, if a vertex is on a domain boundary edge and if an edge has one endpoint on a domain boundary edge, the velocity is only C0$$ {C}^0 $$ at the vertex and on the edge, respectively, and the pressure is totally discontinuous there. Under the condition that no tetrahedron in the grid has more than one face‐triangle on the domain boundary, we prove that the extended finite element is stable, and consequently produces solutions of optimal order convergence for all Stokes problems. A numerical example is given, confirming the theory. [ABSTRACT FROM AUTHOR]

This article introduces a particular weak Galerkin (WG) element on rectangular/cuboid partitions that uses k$$ k $$th order polynomial for weak finite element functions and (k+1)$$ \left(k+1\right) $$th order polynomials for weak derivatives. This WG element is highly accurate with convergence two orders higher than the optimal order in an energy norm and the L2$$ {L}^2 $$ norm. The superconvergence is verified analytically and numerically. Furthermore, the usual stabilizer in the standard weak Galerkin formulation is no longer needed for this element. [ABSTRACT FROM AUTHOR]

The numerical integration for the bilinear form of quadrilateral or hexahedral finite elements can never be exact. It is discovered that the standard Gauss‐Legendre integration is exact if one of the two Qk polynomials in the bilinear form is a Pk polynomial. Based on this observation, it is proved the Qk finite elements, under practical numerical integration, retain the optimal order of convergence on general quadrilateral grids in 2 dimensional space, and on general hexahedral grids in 3 dimensional space. Numerical tests are provided, verifying all theoretic findings. [ABSTRACT FROM AUTHOR]

The weak Galerkin (WG) finite element method is an effective and flexible general numerical technique for solving partial differential equations. A simple WG finite element method is introduced for second‐order elliptic problems. First we have proved that stabilizers are no longer needed for this WG element. Then we have proved the supercloseness of order two for the WG finite element solution. The numerical results confirm the theory. [ABSTRACT FROM AUTHOR]

It is well known that convergence rate of finite element approximation is suboptimal in the L2 norm for solving biharmonic equations when P2 or Q2 element is used. The goal of this paper is to derive a weak Galerkin (WG) P2 element with the L2 optimal convergence rate by assuming the exact solution sufficiently smooth. In addition, our new WG finite element method can be applied to general mesh such as hybrid mesh, polygonal mesh or mesh with hanging node. The numerical experiments have been conducted on different meshes including hybrid meshes with mixed of pentagon and rectangle and mixed of hexagon and triangle. [ABSTRACT FROM AUTHOR]

We propose a local postprocessing method to get a new finite element solution whose flux is conservative element-wise. First, we use the so-called polynomial preserving recovery (postprocessing) technique to obtain a higher order flux which is continuous across the element boundary. Then, we use special bubble functions, which have a nonzero flux only on one face-edge or face-triangle of each element, to correct the finite element solution element by element, guided by the above super-convergent flux and the element mass. The new finite element solution preserves mass element-wise and retains the quasioptimality in approximation. The method produces a conservative flux, of high-order accuracy, satisfying the constitutive law. Numerical tests in 2D and 3D are presented. [ABSTRACT FROM AUTHOR]

In this paper, we revisit the classical error estimates of nonconforming Crouzeix-Raviart type finite elements for the Stokes equations. By introducing some quasi-interpolation operators and using the special properties of these nonconforming elements, it is proved that their consistency errors can be bounded by their approximation errors together with a high-order term, especially which can be of arbitrary order provided that f in the right-hand side is piecewise smooth enough. Furthermore, we show an interesting result that both in the energy norm and L2 norm the consistency errors are dominated by the approximation errors of their finite element spaces. As byproducts, we derive the error estimates in both energy and L2 norms under the regularity assumption ( u, p) ∈ H1 + s(Ω) × H s(Ω) with any s ∈ (0,1], which fills the gap in the a priori error estimate of these nonconforming elements with low regularity [ABSTRACT FROM AUTHOR]

In this article, we propose an iterative method based on the equation decomposition technique (1) for the numerical solution of a singular perturbation problem of fourth-order elliptic equation. At each step of the given method, we only need to solve a boundary value problem of second-order elliptic equation and a second-order singular perturbation problem. We prove that our approximate solution converges to the exact solution when the domain is a disc. Our numerical examples show the efficiency and accuracy of our method. Our iterative method works very well for singular perturbation problems, that is, the case of 0 < ε ≪ 1, and the convergence rate is very fast. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013 [ABSTRACT FROM AUTHOR]

A method of estimating large floods using a linear storage-routing approach is presented. The differences between the proposed approach and those traditionally used are (1) that the flood producing properties of basins are represented by a linear system, (2) the storage parameters of the distributed model are determined using a volume law which, unlike other storage-routing models, accounts for the distribution of storage in natural basins, and (3) the basin outflow hydrograph is determined analytically and expressed in a succinct mathematical form. The single model parameter is estimated from observed data without direct fitting, unlike most traditionally used methods. The model was tested by showing it could reproduce observed large floods on a number of basins. This paper compares the proposed approach with a traditionally used storage routing approach using observed flood data from the Hacking River basin in New South Wales, Australia. Results confirm the usefulness of the proposed approach for estimation of large floods. [ABSTRACT FROM AUTHOR]