共查询到20条相似文献,搜索用时 31 毫秒
1.
Gwanghyun Jo & Do Y. Kwak 《高等学校计算数学学报(英文版)》2020,13(2):281-295
We introduce a low order finite element method for three dimensional elasticity problems. We extend Kouhia-Stenberg element [12] by using two nonconforming components and one conforming component, adding stabilizing terms to the associated bilinear form to ensure the discrete Korn's inequality. Using the second Strang's lemma, we show that our scheme has optimal convergence rates in $L^2$ and piecewise $H^1$-norms even when Poisson ratio $\nu$ approaches $1/2$. Even though some efforts have been made to design a low order method for three dimensional problems in [11,16], their method uses some higher degree basis functions. Our scheme is the first true low order method. We provide three numerical examples which support our analysis. We compute two examples having analytic solutions. We observe the optimal $L^2$ and $H^1$ errors for many different choices of Poisson ratios including the nearly incompressible cases. In the last example, we simulate the driven cavity problem. Our scheme shows non-locking phenomena for the driven cavity problems also. 相似文献
2.
In this paper numerical energy identities of the Yee scheme on uniform grids for three dimensional Maxwell equations with periodic boundary conditions are proposed and expressed in terms of the $L^2$, $H^1$ and $H^2$ norms. The relations between the $H^1$ or $H^2$ semi-norms and the magnitudes of the curls or the second curls of the fields in the Yee scheme are derived. By the $L^2$ form of the identity it is shown that the solution fields of the Yee scheme is approximately energy conserved. By the $H^1$ or $H^2$ semi norm of the identities, it is proved that the curls or the second curls of the solution of the Yee scheme are approximately magnitude (or energy)-conserved. From these numerical energy identities, the Courant-Friedrichs-Lewy (CFL) stability condition is re-derived, and the stability of the Yee scheme in the $L^2$, $H^1$ and $H^2$ norms is then proved. Numerical experiments to compute the numerical energies and convergence orders in the $L^2$, $H^1$ and $H^2$ norms are carried out and the computational results confirm the analysis of the Yee scheme on energy conservation and stability analysis. 相似文献
3.
In this paper, we develop a two-grid method (TGM) based on the FEM for 2D nonlinear time fractional two-term mixed sub-diffusion and diffusion wave equations. A two-grid algorithm is proposed for solving the nonlinear system, which consists of two steps: a nonlinear FE system is solved on a coarse grid, then the linearized FE system is solved on the fine grid by Newton iteration based on the coarse solution. The fully discrete numerical approximation is analyzed, where the Galerkin finite element method for the space derivatives and the finite difference scheme for the time Caputo derivative with order $\alpha\in(1,2)$ and $\alpha_{1}\in(0,1)$. Numerical stability and optimal error estimate $O(h^{r+1}+H^{2r+2}+\tau^{\min\{3-\alpha,2-\alpha_{1}\}})$ in $L^{2}$-norm are presented for two-grid scheme, where $t,$ $H$ and $h$ are the time step size, coarse grid mesh size and fine grid mesh size, respectively. Finally, numerical experiments are provided to confirm our theoretical results and effectiveness of the proposed algorithm. 相似文献
4.
Weak Galerkin finite element methods combined with Crank-Nicolson scheme for parabolic interface problems 下载免费PDF全文
This article is devoted to the a priori error estimates of the fully discrete Crank-Nicolson approximation for the linear parabolic interface problem via weak Galerkin finite element methods (WG-FEM). All the finite element functions are discontinuous for which the usual gradient operator is implemented as distributions in properly defined spaces. Optimal order error estimates in both $L^{\infty}(H^1)$ and $L^{\infty}(L^2)$ norms are established for lowest order WG finite element space $({\cal P}_{k}(K),\;{\cal P}_{k-1}(\partial K),\;\big[{\cal P}_{k-1}(K)\big]^2)$. Finally, we give numerical examples to verify the theoretical results. 相似文献
5.
Kai Wang & Na Wang 《计算数学(英文版)》2022,40(5):777-793
This article concerns numerical approximation of a parabolic interface problem with general $L^2$ initial value. The problem is discretized by a finite element method with a quasi-uniform triangulation of the domain fitting the interface, with piecewise linear approximation to the interface. The semi-discrete finite element problem is furthermore discretized in time by the $k$-step backward difference formula with $ k=1,\ldots,6 $. To maintain high-order convergence in time for possibly nonsmooth $L^2$ initial value, we modify the standard backward difference formula at the first $k-1$ time levels by using a method recently developed for fractional evolution equations. An error bound of $\mathcal{O}(t_n^{-k}\tau^k+t_n^{-1}h^2|\log h|)$ is established for the fully discrete finite element method for general $L^2$ initial data. 相似文献
6.
Weak Galerkin finite element method is introduced for solving wave equation with interface on weak Galerkin finite element space $(\mathcal{P}_k(K), \mathcal{P}_{k−1}(∂K), [\mathcal{P}_{k−1}(K)]^2).$ Optimal order a priori error estimates for both space-discrete scheme and implicit fully discrete scheme are derived in $L^∞(L^2)$ norm. This method uses totally discontinuous functions in approximation space and allows the usage of finite element partitions consisting of general polygonal meshes. Finite element algorithm presented here can contribute to a variety of hyperbolic problems where physical domain consists of heterogeneous media. 相似文献
7.
Ruo Li & Fanyi Yang 《计算数学(英文版)》2023,41(1):39-71
We propose a new least squares finite element method to solve the Stokes problem with two sequential steps. The approximation spaces are constructed by the patch reconstruction with one unknown per element. For the first step, we reconstruct an approximation space consisting of piecewise curl-free polynomials with zero trace. By this space, we minimize a least squares functional to obtain the numerical approximations to the gradient of the velocity and the pressure. In the second step, we minimize another least squares functional to give the solution to the velocity in the reconstructed piecewise divergence-free space. We derive error estimates for all unknowns under both $L^2$ norms and energy norms. Numerical results in two dimensions and three dimensions verify the convergence rates and demonstrate the great flexibility of our method. 相似文献
8.
In this paper, a new discontinuous Galerkin method is developed for
the parabolic equation with jump coefficients satisfying the
continuous flow condition. Theoretical analysis shows that this
method is $L^2$ stable. When the finite element space consists of
interpolative polynomials of degrees $k$, the convergent rate of the
semi-discrete discontinuous Galerkin scheme has an order of$\mathcal{O}(h^k)$. Numerical examples for both 1-dimensional and
2-dimensional problems demonstrate the validity of the new method. 相似文献
9.
Ruo Li & Fanyi Yang 《高等学校计算数学学报(英文版)》2021,14(4):1042-1067
We develop a new least squares method for solving the second-order elliptic equations in non-divergence form. Two least-squares-type functionals are proposed for solving the equation in two sequential steps. We first obtain a numerical
approximation to the gradient in a piecewise irrotational polynomial space. Then
together with the numerical gradient, we seek a numerical solution of the primitive
variable in the continuous Lagrange finite element space. The variational setting
naturally provides an a posteriori error which can be used in an adaptive refinement
algorithm. The error estimates under the $L^2$ norm and the energy norm for both
two unknowns are derived. By a series of numerical experiments, we verify the
convergence rates and show the efficiency of the adaptive algorithm. 相似文献
10.
This is one of our series works on discrete energy analysis of the variable-step BDF schemes. In this part, we present stability and convergence analysis of the third-order BDF (BDF3) schemes with variable steps for linear diffusion equations, see, e.g., [SIAM J. Numer. Anal., 58:2294-2314] and [Math. Comp., 90: 1207-1226] for our previous works on the BDF2 scheme. To this aim, we first build up a discrete gradient structure of the variable-step BDF3 formula under the condition that the adjacent step ratios are less than 1.4877, by which we can establish a discrete energy dissipation law. Mesh-robust stability and convergence analysis in the $L^2$ norm are then obtained. Here the mesh robustness means that the solution errors are well controlled by the maximum time-step size but independent of the adjacent time-step ratios. We also present numerical tests to support our theoretical results. 相似文献
11.
A new second order time stepping ensemble hybridizable discontinuous
Galerkin method for parameterized convection diffusion PDEs with various initial
and boundary conditions, body forces, and time depending coefficients is developed.
For ensemble solutions in $L^∞$($0$, $T$; $L^2$($Ω$)), a superconvergent rate with respect to
the freedom degree of the globally coupled unknowns for all the polynomials of
degree $k$ ≥ $0$ is established. The results of numerical experiments are consistent
with the theoretical findings. 相似文献
12.
Xiang Wang & Yuqing Zhang 《高等学校计算数学学报(英文版)》2021,14(1):47-70
We provide a general construction method for a finite volume element
(FVE) scheme with the optimal $L^2$ convergence rate. The $k$-($k$-1)-order orthogonal
condition (generalized) is proved to be a sufficient and necessary condition for a $k$-order FVE scheme to have the optimal $L^2$ convergence rate in 1D, in which the
independent dual parameters constitute a ($k$-1)-dimension surface in the reasonable
domain in $k$-dimension.In the analysis, the dual strategies in different primary elements are not necessarily to be the same, and they are allowed to be asymmetric in each primary element,
which open up more possibilities of the FVE schemes to be applied to some complex
problems, such as the convection-dominated problems. It worth mentioning that,
the construction can be extended to the quadrilateral meshes in 2D. The stability
and $H^1$ estimate are proved for completeness. All the above results are demonstrated by numerical experiments. 相似文献
13.
As a promising strategy to adjust the order in the variable-order BDF algorithm, a time filtered backward Euler scheme is investigated for the molecular beam
epitaxial equation with slope selection. The temporal second-order convergence in
the $L^2$ norm is established under a convergence-solvability-stability (CSS)-consistent
time-step constraint. The CSS-consistent condition means that the maximum step-size limit required for convergence is of the same order to that for solvability and
stability (in certain norms) as the small interface parameter $ε → 0^+.$ Similar to the
backward Euler scheme, the time filtered backward Euler scheme preserves some
physical properties of the original problem at the discrete levels, including the volume conservation, the energy dissipation law and $L^2$ norm boundedness. Numerical
tests are included to support the theoretical results. 相似文献
14.
In this paper, we discuss the quadrilateral finite element approximation to the two-dimensional linear elasticity problem associated with a homogeneous isotropic elastic material. The optimal convergence of the finite element method is proved for both the $L^2$-norm and energy-norm, and in particular, the convergence is uniform with respect to the Laméconstant $\lambda$. Also the performance of the scheme does not deteriorate as the material becomes nearly incompressible. Numerical experiments are given which are consistent with our theory. 相似文献
15.
Xiu Ye & Shangyou Zhang 《高等学校计算数学学报(英文版)》2023,16(1):230-241
In this paper, we introduce a stabilizer free weak Galerkin (SFWG) finite
element method for second order elliptic problems on rectangular meshes. With
a special weak Gradient space, an order two superconvergence for the SFWG finite
element solution is obtained, in both $L^2$ and $H^1$ norms. A local post-process lifts
such a $P_k$ weak Galerkin solution to an optimal order $P_{k+2}$ solution. The numerical
results confirm the theory. 相似文献
16.
Jin-Feng Wang Min Zhang Hong Li Yang Liu 《Journal of Applied Analysis & Computation》2016,6(2):409-428
In this article, an $H^1$-Galerkin mixed finite element (MFE) method for solving the time fractional water wave model is presented. First-order backward Euler difference method and $L1$ formula are applied to approximate integer derivative and Caputo fractional derivative with order $1/2$, respectively, and $H^1$-Galerkin mixed finite element method is used to approximate the spatial direction. The analysis of stability for fully discrete mixed finite element scheme is made and the optimal space-time orders of convergence for two unknown variables in both $H^1$-norm and $L^2$-norm are derived. Further, some computing results for a priori analysis and numerical figures based on four changed parameters in the studied problem are given to illustrate the effectiveness of the current method 相似文献
17.
Fuzheng Gao & Xiaoshen Wang 《计算数学(英文版)》2015,33(3):307-322
For Sobolev equation, we present a new numerical scheme based on a modified weak Galerkin finite element method, in which differential operators are approximated by weak forms through the usual integration by parts. In particular, the numerical method allows the use of discontinuous finite element functions and arbitrary shape of element. Optimal order error estimates in discrete $H^1$ and $L^2$ norms are established for the corresponding modified weak Galerkin finite element solutions. Finally, some numerical results are given to verify theoretical results. 相似文献
18.
本文针对Brinkman方程引入了一种修正弱Galerkin(MWG)有限元方法.我们通过具有两个离散弱梯度算子的变分形式来逼近模型. 在MWG方法中, 分别用次数为$k$和$k-1$的不连续分段多项式来近似速度函数$u$和压力函数$p$. MWG方法的主要思想是用内部函数的平均值代替边界函数. 因此, 与WG方法相比, MWG方法在不降低准确性的同时, 具有更少的自由度, 对于任意次数不超过$k-1$ 的多项式,MWG方法均可以满足稳定性条件. MWG 方法具有高度的灵活性, 它允许在具有一定形状正则性的任意多边形或多面体上使用不连续函数. 针对$H^1$和$L^22$范数下的速度和压力近似解, 建立了最优阶误差估计. 数值算例表明了该方法的准确性, 收敛性和稳定性. 相似文献
19.
Two-level iteration penalty and variational multiscale method for steady incompressible flows 下载免费PDF全文
In this paper, we study two-level iteration penalty and variational multiscale method for the approximation of steady Navier-Stokes equations at high Reynolds number. Comparing with classical penalty method, this new method does not require very small penalty parameter $\varepsilon$. Moreover, two-level mesh method can save a large amount of CPU time. The error estimates in $H^1$ norm for velocity and in $L^2$ norm for pressure are derived. Finally, two numerical experiments are shown
to support the efficiency of this new method. 相似文献
20.
Let(M~n, g)(n ≥ 3) be an n-dimensional complete Riemannian manifold with harmonic curvature and positive Yamabe constant. Denote by R and R?m the scalar curvature and the trace-free Riemannian curvature tensor of M, respectively. The main result of this paper states that R?m goes to zero uniformly at infinity if for p ≥ n, the L~p-norm of R?m is finite.As applications, we prove that(M~n, g) is compact if the L~p-norm of R?m is finite and R is positive, and(M~n, g) is scalar flat if(M~n, g) is a complete noncompact manifold with nonnegative scalar curvature and finite L~p-norm of R?m. We prove that(M~n, g) is isometric to a spherical space form if for p ≥n/2, the L~p-norm of R?m is sufficiently small and R is positive.In particular, we prove that(M~n, g) is isometric to a spherical space form if for p ≥ n, R is positive and the L~p-norm of R?m is pinched in [0, C), where C is an explicit positive constant depending only on n, p, R and the Yamabe constant. 相似文献