共查询到20条相似文献,搜索用时 15 毫秒
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The interior transmission problem (ITP) plays an important role in the investigation of the inverse scattering problem. In this paper we propose the finite element method for solving the ITP. Based on the $\mathbb T $ -coercivity, we derive both priori error estimate and a posteriori error estimate of the finite element approximation. Numerical experiments are also included to illustrate the accuracy of the finite element method. 相似文献
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A reliable and efficient residual based a posteriori error estimator is constructed for a weakly over-penalized symmetric interior penalty method for second order elliptic problems. Numerical results that demonstrate the performance of the error estimator are presented. 相似文献
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This paper focuses on the optimal error estimates of a linearized semi-implicit scheme for the nematic liquid crystal flows, which is used to describe the time evolution of the materials under the influence of both the flow velocity and the microscopic orientation configurations of rod-like liquid crystal flows. Optimal error estimates of the scheme are proved without any restriction of time step by using an error splitting technique proposed by Li and Sun. Numerical results are provided to confirm the theoretical analysis and the stability of the semi-implicit scheme. 相似文献
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Local Discontinuous Galerkin Finite Element Method and Error Estimates for One Class of Sobolev Equation 总被引:1,自引:0,他引:1
In this paper we present a numerical scheme based on the local discontinuous Galerkin (LDG) finite element method for one
class of Sobolev equations, for example, generalized equal width Burgers equation. The proposed scheme will be proved to have
good numerical stability and high order accuracy for arbitrary nonlinear convection flux, when time variable is continuous.
Also an optimal error estimate is obtained for the fully discrete scheme, when time is discreted by the second order explicit
total variation diminishing (TVD) Runge-Kutta time-marching. Finally some numerical results are given to verify our analysis
for the scheme. 相似文献
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This paper considers the application of the method of boundary penalty terms (SAT) to the numerical solution of the wave equation
on complex shapes with Dirichlet boundary conditions. A theory is developed, in a semi-discrete setting, that allows the use
of a Cartesian grid on complex geometries, yet maintains the order of accuracy with only a linear temporal error-bound. A
numerical example, involving the solution of Maxwell’s equations inside a 2-D circular wave-guide demonstrates the efficacy
of this method in comparison to others (e.g., the staggered Yee scheme)—we achieve a decrease of two orders of magnitude in the level of the L2-error. 相似文献
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In (Xu and Shu in J. Sci. Comput. 40:375–390, 2009), a local discontinuous Galerkin (LDG) method for the surface diffusion of graphs was developed and a rigorous proof for
its energy stability was given. Numerical simulation results showed the optimal order of accuracy. In this subsequent paper,
we concentrate on analyzing a priori error estimates of the LDG method for the surface diffusion of graphs. The main achievement is the derivation of the optimal
convergence rate k+1 in the L
2 norm in one-dimension as well as in multi-dimensions for Cartesian meshes using a completely discontinuous piecewise polynomial
space with degree k≥1. 相似文献
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This paper analyzes the stability and convergence of the Fourier pseudospectral method coupled with a variety of specially designed time-stepping methods of up to fourth order, for the numerical solution of a three dimensional viscous Burgers?? equation. There are three main features to this work. The first is a lemma which provides for an L 2 and H 1 bound on a nonlinear term of polynomial type, despite the presence of aliasing error. The second feature of this work is the development of stable time-stepping methods of up to fourth order for use with pseudospectral approximations of the three dimensional viscous Burgers?? equation. Finally, the main result in this work is that the pseudospectral method coupled with the carefully designed time-discretizations is stable provided only that the time-step and spatial grid-size are bounded by two constants over a finite time. It is notable that this stability condition does not impose a restriction on the time-step that is dependent on the spatial grid size, a fact that is especially useful for three dimensional simulations. 相似文献
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This paper develops an a posteriori error estimate of residual type for finite element approximations of the Allen–Cahn equation ut − Δu+ ε−2 f(u)=0. It is shown that the error depends on ε−1 only in some low polynomial order, instead of exponential order. Based on the proposed a posteriori error estimator, we construct an adaptive algorithm for computing the Allen–Cahn equation and its sharp interface limit, the mean curvature flow. Numerical experiments are also presented to show the robustness and effectiveness of the proposed error estimator and the adaptive algorithm. 相似文献
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Temporal error bounds for the wave equation expressed on second order form are investigated. We show that, with appropriate choices of boundary conditions, the time and space derivatives of the error are bounded even for long times. No long time bound on the error itself is obtained, although numerical experiments indicate that a bound exists. 相似文献
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Thirupathi Gudi Hari Shanker Gupta Neela Nataraj 《Journal of scientific computing》2013,54(1):177-199
Error analysis for a stable C 0 interior penalty method is derived for general fourth order problems on polygonal domains under minimal regularity assumptions on the exact solution. We prove that this method exhibits quasi-optimal order of convergence in the discrete H 2, H 1 and L 2 norms. L ?? norm error estimates are also discussed. Theoretical results are demonstrated by numerical experiments. 相似文献
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A novel discontinuous Galerkin (DG) method is developed to solve time-dependent bi-harmonic type equations involving fourth derivatives in one and multiple space dimensions. We present the spatial DG discretization based on a mixed formulation and central interface numerical fluxes so that the resulting semi-discrete schemes are \(L^2\) stable even without interior penalty. For time discretization, we use Crank–Nicolson so that the resulting scheme is unconditionally stable and second order in time. We present the optimal \(L^2\) error estimate of \(O(h^{k+1})\) for polynomials of degree k for semi-discrete DG schemes, and the \(L^2\) error of \(O(h^{k+1} +(\Delta t)^2)\) for fully discrete DG schemes. Extensions to more general fourth order partial differential equations and cases with non-homogeneous boundary conditions are provided. Numerical results are presented to verify the stability and accuracy of the schemes. Finally, an application to the one-dimensional Swift–Hohenberg equation endowed with a decay free energy is presented. 相似文献
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In this paper, we investigate a characteristic finite element approximation of quadratic optimal control problems governed
by linear advection-dominated diffusion equations, where the state and co-state variables are discretized by piecewise linear
continuous functions and the control variable is approximated by piecewise constant functions. We derive some a priori error
estimates for both the control and state approximations. It is proved that these approximations have convergence order
, where h
U
and h are the spatial mesh-sizes for the control and state discretization, respectively, and k is the time increment. Numerical experiments are presented, which verify the theoretical results.
This research was supported by the National Basic Research Program of China (No. 2007CB814906) and the National Natural Science
Foundation of China (No. 10771124). 相似文献
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Luke Owens 《Journal of scientific computing》2014,59(2):309-333
We analyze an adaptive discontinuous finite element method (ADFEM) for the weakly over-penalized symmetric interior penalty (WOPSIP) operator applied to symmetric positive definite second order elliptic boundary value problems. For first degree polynomials, we prove that the ADFEM is a contraction for the sum of the energy error and the scaled error estimator between two consecutive loops of the adaptive algorithm. After establishing this geometric decay, we define a suitable approximation class and prove that the adaptive WOPSIP method obeys a quasi-optimal rate of convergence. 相似文献
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Following the idea of the tailored finite point method proposed in Han and Huang (J. Comput. Math. 26:728–739, 2008) and Huang (Netw. Heterog. Media 4:91–106, 2009), a series of efficient numerical schemes are developed for the one dimensional scalar wave equation within various types
of media. Stability and accuracy are analyzed and numerically verified. In particular we can obtain unconditionally stable
implicit schemes that can be solved explicitly for boundary value problems. We can also deal with the propagation of discontinuity
and highly oscillatory waves efficiently. The generalization to higher order schemes is straightforward. 相似文献
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In this paper we propose and analyze a Local Discontinuous Galerkin method for an elliptic variational inequality of the first kind that corresponds to a Poisson equation with Signorini type condition on part of the boundary. The method uses piecewise polynomials of degree one for the field variable and of degree zero or one for the approximation of its gradient. We show optimal convergence for the method and illustrate it with some numerical experiments. 相似文献