An Eulerian‐Lagrangian discontinuous Galerkin method for transient advection‐diffusion equations

We develop an Eulerian‐Lagrangian discontinuous Galerkin method for time‐dependent advection‐diffusion equations. The derived scheme has combined advantages of Eulerian‐Lagrangian methods and discontinuous Galerkin methods. The scheme does not contain any undetermined problem‐dependent parameter. An optimal‐order error estimate and superconvergence estimate is derived. Numerical experiments are presented, which verify the theoretical estimates.© 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2007     

Discontinuous Galerkin derivative operators with applications to second‐order elliptic problems and stability

A discontinuous Galerkin (DG) finite‐element interior calculus is used as a common framework to describe various DG approximation methods for second‐order elliptic problems. Using the framework, symmetric interior‐penalty methods, local discontinuous Galerkin methods, and dual‐wind discontinuous Galerkin methods will be compared by expressing all of the methods in primal form. The penalty‐free nature of the dual‐wind discontinuous Galerkin method will be both motivated and used to better understand the analytic properties of the various DG methods. Consideration will be given to Neumann boundary conditions with numerical experiments that support the theoretical results. Many norm equivalencies will be derived laying the foundation for applying dual‐winding techniques to other problems. Copyright © 2015 John Wiley & Sons, Ltd.     

Discontinuous Galerkin finite volume element methods for second‐order linear elliptic problems

In this article, a one parameter family of discontinuous Galerkin finite volume element methods for approximating the solution of a class of second‐order linear elliptic problems is discussed. Optimal error estimates in L² and broken H¹‐ norms are derived. Numerical results confirm the theoretical order of convergences. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

A priori error estimates of an extrapolated space‐time discontinuous galerkin method for nonlinear convection‐diffusion problems

We deal with the numerical solution of a scalar nonstationary nonlinear convection‐diffusion equation. We employ a combination of the discontinuous Galerkin finite element (DGFE) method for the space as well as time discretization. The linear diffusive and penalty terms are treated implicitly whereas the nonlinear convective term is treated by a special higher order explicit extrapolation from the previous time step, which leads to the necessity to solve only a linear algebraic problem at each time step. We analyse this scheme and derive a priori asymptotic error estimates in the L^∞(L²) –norm and the L²(H¹) –seminorm with respect to the mesh size h and time step τ. Finally, we present an efficient solution strategy and numerical examples verifying the theoretical results. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1456–1482, 2010     

On the subdomain‐Galerkin/least squares method for 2‐ and 3‐D mixed elliptic problems with reaction terms

In this article we apply the subdomain‐Galerkin/least squares method, which is first proposed by Chang and Gunzburger for first‐order elliptic systems without reaction terms in the plane, to solve second‐order non‐selfadjoint elliptic problems in two‐ and three‐dimensional bounded domains with triangular or tetrahedral regular triangulations. This method can be viewed as a combination of a direct cell vertex finite volume discretization step and an algebraic least‐squares minimization step in which the pressure is approximated by piecewise linear elements and the flux by the lowest order Raviart‐Thomas space. This combined approach has the advantages of both finite volume and least‐squares methods. Among other things, the combined method is not subject to the Ladyzhenskaya‐Babus?ka‐Brezzi condition, and the resulting linear system is symmetric and positive definite. An optimal error estimate in the H¹(Ω) × H(div; Ω) norm is derived. An equivalent residual‐type a posteriori error estimator is also given. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 738–751, 2002; Published online in Wiley InterScience (www.interscience.wiley.com); DOI 10.1002/num.10030.     

A Petrov–Galerkin spectral method for fourth‐order problems

In this paper, we consider the Petrov–Galerkin spectral method for fourth‐order elliptic problems on rectangular domains subject to non‐homogeneous Dirichlet boundary conditions. We derive some sharp results on the orthogonal approximations in one and two dimensions, which play important roles in numerical solutions of higher‐order problems. By applying these results to a fourth‐order problem, we establish the H²‐error and L²‐error bounds of the Petrov–Galerkin spectral method. Numerical experiments are provided to illustrate the high accuracy of the proposed method and coincide well with the theoretical analysis. Copyright © 2013 John Wiley & Sons, Ltd.     

A Crank‐Nicolson Petrov‐Galerkin method with quadrature for semi‐linear parabolic problems

We propose and analyze an application of a fully discrete C² spline quadrature Petrov‐Galerkin method for spatial discretization of semi‐linear parabolic initial‐boundary value problems on rectangular domains. We prove second order in time and optimal order H¹ norm convergence in space for the extrapolated Crank‐Nicolson quadrature Petrov‐Galerkin scheme. We demonstrate numerically both L² and H¹ norm optimal order convergence of the scheme even if the nonlinear source term is not smooth. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005.     

Stability and convergence of two‐grid Crank‐Nicolson extrapolation scheme for the time‐dependent natural convection equations

In this paper, we consider the Crank‐Nicolson extrapolation scheme for the 2D/3D unsteady natural convection problem. Our numerical scheme includes the implicit Crank‐Nicolson scheme for linear terms and the recursive linear method for nonlinear terms. Standard Galerkin finite element method is used to approximate the spatial discretization. Stability and optimal error estimates are provided for the numerical solutions. Furthermore, a fully discrete two‐grid Crank‐Nicolson extrapolation scheme is developed, the corresponding stability and convergence results are derived for the approximate solutions. Comparison from aspects of the theoretical results and computational efficiency, the two‐grid Crank‐Nicolson extrapolation scheme has the same order as the one grid method for velocity and temperature in H¹‐norm and for pressure in L²‐norm. However, the two‐grid scheme involves much less work than one grid method. Finally, some numerical examples are provided to verify the established theoretical results and illustrate the performances of the developed numerical schemes.     

One‐level and two‐level finite element methods for the Boussinesq problem

Previous works on the convergence of numerical methods for the Boussinesq problem were conducted, while the optimal L²‐norm error estimates for the velocity and temperature are still lacked. In this paper, the backward Euler scheme is used to discrete the time terms, standard Galerkin finite element method is adopted to approximate the variables. The MINI element is used to approximate the velocity and pressure, the temperature field is simulated by the linear polynomial. Under some restriction on the time step, we firstly present the optimal L² error estimates of approximate solutions. Secondly, two‐level method based on Stokes iteration for the Boussinesq problem is developed and the corresponding convergence results are presented. By this method, the original problem is decoupled into two small linear subproblems. Compared with the standard Galerkin method, the two‐level method not only keeps good accuracy but also saves a lot of computational cost. Finally, some numerical examples are provided to support the established theoretical analysis.     

Weak Galerkin finite element methods for Parabolic equations

A newly developed weak Galerkin method is proposed to solve parabolic equations. This method allows the usage of totally discontinuous functions in approximation space and preserves the energy conservation law. Both continuous and discontinuous time weak Galerkin finite element schemes are developed and analyzed. Optimal‐order error estimates in both H¹ and L² norms are established. Numerical tests are performed and reported. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

An optimal‐order error estimate on an H1‐Galerkin mixed method for a nonlinear parabolic equation in porous medium flow

We present an H¹‐Galerkin mixed finite element method for a nonlinear parabolic equation, which models a compressible fluid flow process in subsurface porous media. The method possesses the advantages of mixed finite element methods while avoiding directly inverting the permeability tensor, which is important especially in a low permeability zone. We conducted theoretical analysis to study the existence and uniqueness of the numerical solutions of the scheme and prove an optimal‐order error estimate for the method. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

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.     

Chebyshev‐Legendre spectral method for solving the two‐dimensional vorticity equations with homogeneous Dirichlet conditions

The Chebyshev‐Legendre spectral method for the two‐dimensional vorticity equations is considered. The Legendre Galerkin Chebyshev collocation method is used with the Chebyshev‐Gauss collocation points. The numerical analysis results under the L²‐norm for the Chebyshev‐Legendre method of one‐dimensional case are generalized into that of the two‐dimensional case. The stability and optimal order convergence of the method are proved. Numerical results are given. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

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.     

Methods for the numerical solution of the Benjamin‐Bona‐Mahony‐Burgers equation

L^∞‐error estimates for finite element for Galerkin solutions for the Benjamin‐Bona‐Mahony‐Burgers (BBMB) equation are considered. A priori bound and the semidiscrete Galerkin scheme are studied using appropriate projections. For fully discrete Galerkin schemes, we consider the backward Euler method and analyze the corresponding error estimates. For a second order accuracy in time, we propose a three‐level backward method. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

Convergence of a Galerkin method for 2‐D discontinuous Euler flows

We prove the convergence of a discontinuous Galerkin method approximating the 2‐D incompressible Euler equations with discontinuous initial vorticity: ω₀ ϵ L²(Ω). Furthermore, when ω₀ ϵ L^∞(Ω), the whole sequence is shown to be strongly convergent. This is the first convergence result in numerical approximations of this general class of discontinuous flows. Some important flows such as vortex patches belong to this class. © 2000 John Wiley & Sons, Inc.     

Combination of standard Galerkin and subspace methods for the time‐dependent Navier‐Stokes equations with nonsmooth initial data

We consider a combination of the standard Galerkin method and the subspace decomposition methods for the numerical solution of the two‐dimensional time‐dependent incompressible Navier‐Stokes equations with nonsmooth initial data. Because of the poor smoothness of the solution near t = 0, we use the standard Galerkin method for time interval [0, 1] and the subspace decomposition method time interval [1, ∞). The subspace decomposition method is based on the solution into the sum of a low frequency component integrated using a small time step Δt and a high frequency integrated using a larger time step pΔt with p > 1. From the H¹‐stability and L²‐error analysis, we show that the subspace decomposition method can yield a significant gain in computing time. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2009     

A high‐order B‐spline collocation scheme for solving a nonhomogeneous time‐fractional diffusion equation

A high‐accuracy numerical approach for a nonhomogeneous time‐fractional diffusion equation with Neumann and Dirichlet boundary conditions is described in this paper. The time‐fractional derivative is described in the sense of Riemann‐Liouville and discretized by the backward Euler scheme. A fourth‐order optimal cubic B‐spline collocation (OCBSC) method is used to discretize the space variable. The stability analysis with respect to time discretization is carried out, and it is shown that the method is unconditionally stable. Convergence analysis of the method is performed. Two numerical examples are considered to demonstrate the performance of the method and validate the theoretical results. It is shown that the proposed method is of order O(Δx⁴ + Δt^{2 ? α}) convergence, where α ∈ (0,1) . Moreover, the impact of fractional‐order derivative on the solution profile is investigated. Numerical results obtained by the present method are compared with those obtained by the method based on standard cubic B‐spline collocation method. The CPU time for present numerical method and the method based on cubic B‐spline collocation method are provided.     

Local discontinuous Galerkin methods with explicit Runge‐Kutta time marching for nonlinear carburizing model

A fully discrete local discontinuous Galerkin (LDG) method coupled with 3 total variation diminishing Runge‐Kutta time‐marching schemes, for solving a nonlinear carburizing model, will be analyzed and implemented in this paper. On the basis of a suitable numerical flux setting in the LDG method, we obtain the optimal error estimate for the Runge‐Kutta–LDG schemes by energy analysis, under the condition τ ≤ λh², where h and τ are mesh size and time step, respectively, λ is a positive constant independent of h. Numerical experiments are presented to verify the accuracy and capability of the proposed schemes. For the carburizing diffusion processes of steel and the diffusion simulation for Cu‐Ni system, the numerical results show good agreement with the experimental results.     

Local discontinuous galerkin approximation of convection‐dominated diffusion optimal control problems with control constraints

In this article, we investigate local discontinuous Galerkin approximation of stationary convection‐dominated diffusion optimal control problems with distributed control constraints. The state variable and adjoint state variable are approximated by piecewise linear polynomials without continuity requirement, whereas the control variable is discretized by variational discretization concept. The discrete first‐order optimality condition is derived. We show that optimization and discretization are commutative for the local discontinuous Galerkin approximation. Because the solutions to convection‐dominated diffusion equations often admit interior or boundary layers, residual type a posteriori error estimate in L² norm is proved, which can be used to guide mesh refinement. Finally, numerical examples are presented to illustrate the theoretical findings. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 339–360, 2014