Multiscale discontinuous Petrov–Galerkin method for the multiscale elliptic problems

In this article, we present a new multiscale discontinuous Petrov–Galerkin method (MsDPGM) for multiscale elliptic problems. This method utilizes the classical oversampling multiscale basis in the framework of a Petrov–Galerkin version of the discontinuous Galerkin method, allowing us to better cope with multiscale features in the solution. MsDPGM takes advantage of the multiscale Petrov–Galerkin method (MsPGM) and the discontinuous Galerkin method (DGM). It can eliminate the resonance error completely and decrease the computational costs of assembling the stiffness matrix, thus, allowing for more efficient solution algorithms. On the basis of a new H² norm error estimate between the multiscale solution and the homogenized solution with the first‐order corrector, we give a detailed convergence analysis of the MsDPGM under the assumption of periodic oscillating coefficients. We also investigate a multiscale discontinuous Galerkin method (MsDGM) whose bilinear form is the same as that of the DGM but the approximation space is constructed from the classical oversampling multiscale basis functions. This method has not been analyzed theoretically or numerically in the literature yet. Numerical experiments are carried out on the multiscale elliptic problems with periodic and randomly generated log‐normal coefficients. Their results demonstrate the efficiency of the proposed method.     

The local discontinuous Galerkin method for the Oseen equations

We introduce and analyze the local discontinuous Galerkin method for the Oseen equations of incompressible fluid flow. For a class of shape-regular meshes with hanging nodes, we derive optimal a priori estimates for the errors in the velocity and the pressure in - and negative-order norms. Numerical experiments are presented which verify these theoretical results and show that the method performs well for a wide range of Reynolds numbers.

     

Technical Note: A note on the selection of the penalty parameter for discontinuous Galerkin finite element schemes

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     

On discontinuous Galerkin method and semiregular family of triangulations

Discretization of second order elliptic partial differential equations by discontinuous Galerkin method often results in numerical schemes with penalties. In this paper we analyze these penalized schemes in the context of quite general triangular meshes satisfying only a semiregularity assumption. A new (modified) penalty term is presented and theoretical properties are proven together with illustrative numerical results. This work is a part of the research project MSM 0021620839 financed by MSMT and was partly supported by the project No. 201/04/1503 of the Grant Agency of the Czech Republic.     

The discontinuous Galerkin method for discretely observed Asian options

Asian options represent an important subclass of the path-dependent contracts that are identified by payoff depending on the average of the underlying asset prices over the prespecified period of option lifetime. Commonly, this average is observed at discrete dates, and also, early exercise features can be admitted. As a result, analytical pricing formulae are not always available. Therefore, some form of a numerical approximation is essential for efficient option valuation. In this paper, we study a PDE model for pricing discretely observed arithmetic Asian options with fixed as well as floating strike for both European and American exercise features. The pricing equation for such options is similar to the Black-Scholes equation with 1 underlying asset, and the corresponding average appears only in the jump conditions across the sampling dates. The objective of the paper is to present the comprehensive methodological concept that forms and improves the valuation process. We employ a robust numerical procedure based on the discontinuous Galerkin approach arising from the piecewise polynomial generally discontinuous approximations. This technique enables a simple treatment of discrete sampling by incorporation of jump conditions at each monitoring date. Moreover, an American early exercise constraint is directly handled as an additional nonlinear source term in the pricing equation. The proposed solving procedure is accompanied by an empirical study with practical results compared to reference values.     

Analysis of a fully discrete local discontinuous Galerkin method for time-fractional fourth-order problems

In this paper, we propose and analyze a fully discrete local discontinuous Galerkin (LDG) finite element method for time-fractional fourth-order problems. The method is based on a finite difference scheme in time and local discontinuous Galerkin methods in space. Stability is ensured by a careful choice of interface numerical fluxes. We prove that our scheme is unconditional stable and convergent. Numerical examples are shown to illustrate the efficiency and accuracy of our scheme.     

A selective immersed discontinuous Galerkin method for elliptic interface problems

This article proposes a selective immersed discontinuous Galerkin method based on bilinear immersed finite elements (IFE) for solving second‐order elliptic interface problems. This method applies the discontinuous Galerkin formulation wherever selected, such as those elements around an interface or a singular source, but the regular Galerkin formulation everywhere else. A selective bilinear IFE space is constructed and applied to the selective immersed discontinuous Galerkin method based on either the symmetric or nonsymmetric interior penalty discontinuous Galerkin formulation. The new method can solve an interface problem by a rectangular mesh with local mesh refinement independent of the interface even if its geometry is nontrivial. Meanwhile, if desired, its computational cost can be maintained very close to that of the standard Galerkin IFE method. It is shown that the selective bilinear IFE space has the optimal approximation capability expected from piecewise bilinear polynomials. Numerical examples are provided to demonstrate features of this method, including the effectiveness of local mesh refinement around the interface and the sensitivity to the penalty parameters. Copyright © 2013 John Wiley & Sons, Ltd.     

一种推广的对流扩散方程的局部化间断Galerkin方法

研究求解一种产生于径向渗流问题的推广的对流扩散方程的局部化间断Galerkin方法，对一般非线性情形证明了方法的L^2稳定性；对线性情形证明了，当方法取有限元空间为κ次多项式空间时，数值解逼近的L^∞（0，T；L^2)模的误差阶为κ。     

Analysis and convergence of finite volume method using discontinuous bilinear functions

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     

Optimal error estimates of the local discontinuous Galerkin method for nonlinear second‐order elliptic problems on Cartesian grids

In this paper, we study the local discontinuous Galerkin (LDG) methods for two‐dimensional nonlinear second‐order elliptic problems of the type u_xx + u_yy = f(x, y, u, u_x, u_y) , in a rectangular region Ω with classical boundary conditions on the boundary of Ω . Convergence properties for the solution and for the auxiliary variable that approximates its gradient are established. More specifically, we use the duality argument to prove that the errors between the LDG solutions and the exact solutions in the L² norm achieve optimal (p + 1)th order convergence, when tensor product polynomials of degree at most p are used. Moreover, we prove that the gradient of the LDG solution is superclose with order p + 1 toward the gradient of Gauss–Radau projection of the exact solution. The results are valid in two space dimensions on Cartesian meshes using tensor product polynomials of degree p ≥ 1 , and for both mixed Dirichlet–Neumann and periodic boundary conditions. Preliminary numerical experiments indicate that our theoretical findings are optimal.     

Optimal error estimates for the hp-version interior penalty discontinuous Galerkin finite element method

We consider the hp-version interior penalty discontinuous Galerkinfinite-element method (hp-DGFEM) for second-order linear reaction–diffusionequations. To the best of our knowledge, the sharpest knownerror bounds for the hp-DGFEM are due to Rivière et al.(1999,Comput. Geosci., 3, 337–360) and Houston et al.(2002,SIAM J. Numer. Anal., 99, 2133–2163). These are optimalwith respect to the meshsize h but suboptimal with respect tothe polynomial degree p by half an order of p. We present improvederror bounds in the energy norm, by introducing a new functionspace framework. More specifically, assuming that the solutionsbelong element-wise to an augmented Sobolev space, we deducefully hp-optimal error bounds.     

Convergence order estimates of the local discontinuous Galerkin method for instationary Darcy flow

We present a new a priori stability and convergence analysis for the local discontinuous Galerkin method applied to the instationary Darcy problem formulated on a d‐dimensional polytope with nonhomogeneous Neumann and Dirichlet boundary conditions. In addition to including a spatially and temporally varying permeability tensor into all estimates, the utilized analysis technique produces a convergence order result for the primary and the flux variables. The only stabilization in the proposed scheme is represented by a penalty term in the primary unknown, and our analysis provides some insights into the role played by this particular choice of stabilization. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1374–1394, 2017     

A posteriori error estimate for discontinuous Galerkin finite element method on polytopal mesh

In this work, we derive a posteriori error estimates for discontinuous Galerkin finite element method on polytopal mesh. We construct a reliable and efficient a posteriori error estimator on general polygonal or polyhedral meshes. An adaptive algorithm based on the error estimator and DG method is proposed to solve a variety of test problems. Numerical experiments are performed to illustrate the effectiveness of the algorithm.     

Analysis of a combined mixed finite element and discontinuous Galerkin method for incompressible two‐phase flow in porous media

We analyze a combined method consisting of the mixed finite element method for pressure equation and the discontinuous Galerkin method for saturation equation for the coupled system of incompressible two‐phase flow in porous media. The existence and uniqueness of numerical solutions are established under proper conditions by using a constructive approach. Optimal error estimates in L²(H¹) for saturation and in L ^∞ (H(div)) for velocity are derived. Copyright © 2013 John Wiley & Sons, Ltd.     

Local discontinuous Galerkin method for solving an N‐carrier system

In this article, we present local discontinuous Galerkin (LDG) method for solving a model of energy exchanges in an N ‐carrier system with Neumann boundary conditions. This model extends the concept of the well‐known parabolic two‐step model for microheat transfer to the energy exchanges in a generalized N ‐carrier system with heat sources. The energy norm stability and error estimate of the LDG method is proved for solving N ‐carrier system. Some numerical examples are given. The numerical results when compared with the exact solution and other numerical results indicate that the present method is seen to be a very good alternative to some existing techniques for realistic problems. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2011     

Wavelets and adaptive grids for the discontinuous Galerkin method

In this paper, space adaptivity is introduced to control the error in the numerical solution of hyperbolic systems of conservation laws. The reference numerical scheme is a new version of the discontinuous Galerkin method, which uses an implicit diffusive term in the direction of the streamlines, for stability purposes. The decision whether to refine or to unrefine the grid in a certain location is taken according to the magnitude of wavelet coefficients, which are indicators of local smoothness of the numerical solution. Numerical solutions of the nonlinear Euler equations illustrate the efficiency of the method.     

A posteriori error estimates for discontinuous Galerkin method to the elasticity problem

This work concerns with the discontinuous Galerkin (DG) method for the time‐dependent linear elasticity problem. We derive the a posteriori error bounds for semidiscrete and fully discrete problems, by making use of the stationary elasticity reconstruction technique which allows to estimate the error for time‐dependent problem through the error estimation of the associated stationary elasticity problem. For fully discrete scheme, we make use of the backward‐Euler scheme and an appropriate space‐time reconstruction. The technique here can be applicable for a variety of DG methods as well.     

Local and pointwise error estimates of the local discontinuous Galerkin method applied to the Stokes problem

We prove local and pointwise error estimates for the local discontinuous Galerkin method applied to the Stokes problem in two and three dimensions. By using techniques originally developed by A. Schatz [Math. Comp., 67 (1998), 877-899] to prove pointwise estimates for the Laplace equation, we prove optimal weighted pointwise estimates for both the velocity and the pressure for domains with smooth boundaries.

     

Interior penalty discontinuous Galerkin methods with implicit time‐integration techniques for nonlinear parabolic equations

We prove existence and numerical stability of numerical solutions of three fully discrete interior penalty discontinuous Galerkin methods for solving nonlinear parabolic equations. Under some appropriate regularity conditions, we give the l²(H¹) and l^∞(L²) error estimates of the fully discrete symmetric interior penalty discontinuous Galerkin–scheme with the implicit θ ‐schemes in time, which include backward Euler and Crank–Nicolson finite difference approximations. Our estimates are optimal with respect to the mesh size h. The theoretical results are confirmed by some numerical experiments. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Time discontinuous Galerkin space-time finite element method for nonlinear Sobolev equations

This article presents a complete discretization of a nonlinear Sobolev equation using space-time discontinuous Galerkin method that is discontinuous in time and continuous in space. The scheme is formulated by introducing the equivalent integral equation of the primal equation. The proposed scheme does not explicitly include the jump terms in time, which represent the discontinuity characteristics of approximate solution. And then the complexity of the theoretical analysis is reduced. The existence and uniqueness of the approximate solution and the stability of the scheme are proved. The optimalorder error estimates in L²(H¹) and L²(L²) norms are derived. These estimates are valid under weak restrictions on the space-time mesh, namely, without the condition k_n≥ch², which is necessary in traditional space-time discontinuous Galerkin methods. Numerical experiments are presented to verify the theoretical results.