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1.
I. Herrera 《Numerical Methods for Partial Differential Equations》2007,23(3):597-639
A truly general and systematic theory of finite element methods (FEM) should be formulated using, as trial and test functions, piecewise‐defined functions that can be fully discontinuous across the internal boundary, which separates the elements from each other. Some of the most relevant work addressing such formulations is contained in the literature on discontinuous Galerkin (dG) methods and on Trefftz methods. However, the formulations of partial differential equations in discontinuous functions used in both of those fields are indirect approaches, which are based on the use of Lagrange multipliers and mixed methods, in the case of dG methods, and the frame, in the case of Trefftz method. This article addresses this problem from a different point of view and proposes a theory, formulated in discontinuous piecewise‐defined functions, which is direct and systematic, and furthermore it avoids the use of Lagrange multipliers or a frame, while mixed methods are incorporated as particular cases of more general results implied by the theory. When boundary value problems are formulated in discontinuous functions, well‐posed problems are boundary value problems with prescribed jumps (BVPJ), in which the boundary conditions are complemented by suitable jump conditions to be satisfied across the internal boundary of the domain‐partition. One result that is presented in this article shows that for elliptic equations of order 2m, with m ≥ 1, the problem of establishing conditions for existence of solution for the BVPJ reduces to that of the “standard boundary value problem,” without jumps, which has been extensively studied. Actually, this result is an illustration of a more general one that shows that the same happens for any differential equation, or system of such equations that is linear, independently of its type and with possibly discontinuous coefficients. This generality is achieved by means of an algebraic framework previously developed by the author and his collaborators. A fundamental ingredient of this algebraic formulation is a kind of Green's formulas that simplify many problems (some times referred to as Green‐Herrera formulas). An important practical implication of our approach is worth mentioning: “avoiding the introduction of the Lagrange multipliers, or the ‘frame’ in the case of Trefftz‐methods, significantly reduces the number of degrees of freedom to be dealt with.” © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007 相似文献
2.
Mark Ainsworth Richard Rankin 《Numerical Methods for Partial Differential Equations》2012,28(3):1099-1104
We obtain a computable lower bound on the value of the interior penalty parameters sufficient for the existence of a unique discontinuous Galerkin finite element approximation of a second order elliptic problem. The bound obtained is valid for meshes containing an arbitrary number of hanging nodes and elements of arbitrary nonuniform polynomial order. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011 相似文献
3.
Yingjie Liu;Chi-Wang Shu;Eitan Tadmor;Mengping Zhang 《Mathematical Modelling and Numerical Analysis》2011,45(6):1009-1032
In this paper we present two versions of the central localdiscontinuous Galerkin (LDG) method on overlapping cellsfor solving diffusion equations, and provide theirstability analysis and error estimates for the linear heat equation.A comparisonbetween the traditional LDG method ona single mesh and the two versions of the central LDGmethod on overlapping cells is also made.Numerical experiments are provided to validate the quantitativeconclusions from the analysis and to support conclusions forgeneral polynomial degrees.https://doi.org/10.1051/m2an/2011007 相似文献
4.
In [35, 36], we presented an $h$-adaptive Runge-Kutta
discontinuous Galerkin method using troubled-cell indicators for
solving hyperbolic conservation laws. A tree data structure (binary
tree in one dimension and quadtree in two dimensions) is used to aid
storage and neighbor finding. Mesh adaptation is achieved by
refining the troubled cells and coarsening the untroubled
"children". Extensive numerical tests indicate that the proposed $h$-adaptive method is capable of saving the computational cost and
enhancing the resolution near the discontinuities. In this paper, we
apply this $h$-adaptive method to solve Hamilton-Jacobi equations,
with an objective of enhancing the resolution near the
discontinuities of the solution derivatives. One- and
two-dimensional numerical examples are shown to illustrate the
capability of the method. 相似文献
5.
In this paper, we present further development of the local discontinuous Galerkin （LDG） method designed in [21] and a new dissipative discontinuous Galerkin （DG） method for the HuntermSaxton equation. The numerical fluxes for the LDG and DG methods in this paper are based on the upwinding principle. The resulting schemes provide additional energy dissipation and better control of numerical oscillations near derivative singularities. Stability and convergence of the schemes are proved theoretically, and numerical simulation results are provided to compare with the scheme in [21]. 相似文献
6.
Discontinuous Stable Elements for the Incompressible Flow 总被引：4，自引：0，他引：4
Xiu Ye 《Advances in Computational Mathematics》2004,20(4):333-345
In this paper, we derive a discontinuous Galerkin finite element formulation for the Stokes equations and a group of stable elements associated with the formulation. We prove that these elements satisfy the new inf–sup condition and can be used to solve incompressible flow problems. Associated with these stable elements, optimal error estimates for the approximation of both velocity and pressure in L
2 norm are obtained for the Stokes problems, as well as an optimal error estimate for the approximation of velocity in a mesh dependent norm. 相似文献
7.
In this paper, we investigate a fully discrete local discontinuous Galerkin approximation of a non-linear non-Fickian diffusion model in viscoelastic polymers. For the spatial discretization, we adopt local discontinuous Galerkin finite element method and for the time discretization we use backward Euler method. We derive the stability estimate and a priori error estimate for the discrete scheme. Numerical examples are given to verify the theoretical findings. 相似文献
8.
Xiu Ye 《Numerical Methods for Partial Differential Equations》2008,24(1):335-348
We develop finite volume method using discontinuous bilinear functions on rectangular mesh. This method is analyzed for the Stokes equations. An optimal error estimate for the approximation of velocity is obtained in a mesh‐dependent norm. First order L2‐error estimates are derived for the approximations of both velocity and pressure. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007 相似文献
9.
10.
In this article, we prove some weighted pointwise estimates for three discontinuous Galerkin methods with lifting operators appearing in their corresponding bilinear forms. We consider a Dirichlet problem with a general second-order elliptic operator.
11.
S. V. Khabirov 《Siberian Mathematical Journal》2002,43(5):942-954
We consider three-dimensional subalgebras admitted by the equations of gas dynamics having time as an invariant and containing no rotation operator. For such subalgebras we seek for irregular partially invariant solutions of rank 2 and defect 1. The representation for solutions has the form which generalizes motion of a gas with a linear velocity field. We show that partially invariant solutions exist for each subalgebra. We describe the set of these solutions. We find solutions with the indicated representation that are not partially invariant. The solutions reducible to invariant solutions are generalized to new submodels. 相似文献
12.
双曲守恒律的几种新数值方法的比较研究 总被引：5，自引：0，他引：5
本文就一维线性双曲方程的光滑和间断两种初值问题的求解,对双曲守恒律的三种新数值方法,即,WENO方法、间断Galerkin方法和全局复合方法,进行了数值比较实验,在精度、计算速度等方面的比较上,对这三个方法有了一个较详细的了解,得到了一些有用的结论。 相似文献
13.
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
14.
Yongbin Han;Yanren Hou 《Mathematical Modelling and Numerical Analysis》2021,55(5):2349-2364
In this paper,the a priori error estimates of an embedded discontinuous Galerkin method for the Oseen equations are presented.Please check whether short title on odd pages have been set correctly. It is proved that the velocity error in the L 2(Ω) norm,has an optimal error bound with convergence order k +1,where the constants are dependent on the Reynolds number (or ν −1),in the diffusion-dominated regime,and in the convection-dominated regime,it has a Reynolds-robust error bound with quasi-optimal convergence order k +1/2. Here,k is the polynomial order of the velocity space.Please provide missing AMS classification codes. In addition,we also prove an optimal error estimate for the pressure. Finally,we carry out some numerical experiments to corroborate our analytical results.https://doi.org/10.1051/m2an/2021059 相似文献
15.
The authors study the multi-soliton,multi-cuspon solutions to the CamassaHolm equation and their interaction.According to the solution formula due to Li in 2004and 2005,the authors give the proper choi... 相似文献
16.
Jing Wen Jian Su Yinnian He Hongbin Chen 《Numerical Methods for Partial Differential Equations》2021,37(1):383-405
In this paper, a semi‐discrete scheme and a fully discrete scheme of the Stokes‐Biot model are proposed, and we analyze the semi‐discrete scheme in detail. First of all, we prove the existence and uniqueness of the semi‐discrete scheme, and a‐priori error estimates are derived. Then, we present the same conclusions for the fully discrete scheme. Finally, under both matching and non‐matching meshes some numerical tests are given to validate the analysis of convergence, which well support the theoretical results. 相似文献
17.
Piotr Krzyżanowski 《Numerical Methods for Partial Differential Equations》2016,32(6):1572-1590
The condition number of a discontinuous Galerkin finite element discretization preconditioned with a nonoverlapping additive Schwarz method is analyzed. We improve the result of Antonietti and Houston (J Sci Comput 46 (2011), 124–149), where a bound has been proved for a two‐level nonoverlapping additive Schwarz method with coarse problem using polynomials of degree on a coarse mesh size . In a more general framework, where the concurrency of the algorithm is increased by applying solvers on subdomains smaller than the coarse grid cells, we prove that the condition number of the preconditioned system is where is the coarse space element degree polynomial and is the size of subdomain where local problems are solved in parallel. Our result also extends to the case of discontinuous coefficient, piecewise constant on the coarse grid, for a composite continuous–discontinuous Galerkin discretization. © 2016Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1572–1590, 2016 相似文献
18.
This article presents a new type of second‐order scheme for solving the system of Euler equations, which combines the Runge‐Kutta discontinuous Galerkin (DG) finite element method and the kinetic flux vector splitting (KFVS) scheme. We first discretize the Euler equations in space with the DG method and then the resulting system from the method‐of‐lines approach will be discretized using a Runge‐Kutta method. Finally, a second‐order KFVS method is used to construct the numerical flux. The proposed scheme preserves the main advantages of the DG finite element method including its flexibility in handling irregular solution domains and in parallelization. The efficiency and effectiveness of the proposed method are illustrated by several numerical examples in one‐ and two‐dimensional spaces. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006 相似文献
19.
Clint Dawson Jennifer Proft 《Numerical Methods for Partial Differential Equations》2001,17(6):545-564
The local discontinuous Galerkin method has been developed recently by Cockburn and Shu for convection‐dominated convection‐diffusion equations. In this article, we consider versions of this method with interior penalties for the numerical solution of transport equations, and derive a priori error estimates. We consider two interior penalty methods, one that penalizes jumps in the solution across interelement boundaries, and another that also penalizes jumps in the diffusive flux across such boundaries. For the first penalty method, we demonstrate convergence of order k in the L∞(L2) norm when polynomials of minimal degree k are used, and for the second penalty method, we demonstrate convergence of order k+1/2. Through a parabolic lift argument, we show improved convergence of order k+1/2 (k+1) in the L2(L2) norm for the first penalty method with a penalty parameter of order one (h?1). © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 545–564, 2001 相似文献
20.
Ismael Herrera‐Revilla 《Numerical Methods for Partial Differential Equations》2008,24(3):845-878
This article is devoted to introduce a new approach to iterative substructuring methods that, without recourse to Lagrange multipliers, yields positive definite preconditioned formulations of the Neumann–Neumann and FETI types. To my knowledge, this is the first time that such formulations have been made without resource to Lagrange multipliers. A numerical advantage that is concomitant to such multipliers‐free formulations is the reduction of the degrees of freedom associated with the Lagrange multipliers. Other attractive features are their generality, directness, and simplicity. The general framework of the new approach is rather simple and stems directly from the discretization procedures that are applied; in it, the differential operators act on discontinuous piecewise‐defined functions. Then, the Lagrange multipliers are not required because in such an environment the functions‐discontinuities are not an anomaly that need to be corrected. The resulting algorithms and equations‐systems are also derived with considerable detail. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008 相似文献