Stokes' first problem for a Newtonian fluid in a non‐Darcian porous half‐space using a Laguerre–Galerkin method

A Laguerre–Galerkin method is proposed and analysed for the Stokes' first problem of a Newtonian fluid in a non‐Darcian porous half‐space on a semi‐infinite interval. It is well known that Stokes' first problem has a jump discontinuity on boundary which is the main obstacle in numerical methods. By reformulating this equation with suitable functional transforms, it is shown that the Laguerre–Galerkin approximations are convergent on a semi‐infinite interval with spectral accuracy. An efficient and accurate algorithm based on the Laguerre–Galerkin approximations of the transformed equations is developed and implemented. Numerical results indicating the high accuracy and effectiveness of this algorithm are presented. Copyright © 2007 John Wiley & Sons, Ltd.     

Convergence analysis of a coupled method for time‐dependent convection‐diffusion equations

The coupling of two locally mass conservative methods is formulated and analyzed for the time‐dependent convection‐diffusion problem. Finite volume method is used in some subdomains and interior penalty discontinuous Galerkin method is used in other subdomains. Numerical examples show the advantages of the proposed hybrid method, namely an accurate approximation obtained at a reduced computational cost. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 133–157, 2014     

Approximation of solutions to operator equations in spaces of smooth functions and in spaces of distributions

The Galerkin method, in particular, the Galerkin method with finite elements (called finite element method) is widely used for numerical solution of differential equations. The Galerkin method allows us to obtain approximations of weak solutions only. However, there arises in applications a rich variety of problems where approximations of smooth solutions and solutions in the sense of distributions have to be found. This article is devoted to the employment of the Petrov–Galerkin method for solving such problems. The article contains general results on the Petrov–Galerkin approximations of solutions to linear and nonlinear operator equations. The problem on construction of the subspaces, which ensure the convergence of the approximations, is investigated. We apply the general results to two‐dimensional (2D) and 3D problems of the elasticity, to a parabolic problem, and to a nonlinear problem of the plasticity. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 406–450, 2014     

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.     

On a boundary variational inequality of the second kind modelling a friction problem

A scalar contact problem with friction governed by the Yukawa equation is reduced to a boundary variational inequality. The presence of the non‐differentiable friction functional causes some difficulties when approximated. We present two methods to overcome this difficulty. The first one is a regularization leading to a non‐linear boundary variational equation, for which we propose an iterative procedure, whereas the second method is based on the boundary mixed variational formulation involving Lagrange multipliers. We propose Uzawa's algorithm to compute the saddle point of the corresponding boundary Lagrangian and investigate the discretization of various formulations by the boundary element Galerkin method. Convergence of the boundary element solution is proved and a convergence order is obtained. Copyright © 2002 John Wiley & Sons, Ltd.     

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.     

Superconvergence analysis of the Galerkin FEM for a singularly perturbed convection–diffusion problem with characteristic layers

We analyze the superconvergence property of the Galerkin finite element method (FEM) for elliptic convection–diffusion problems with characteristic layers. This method on Shishkin meshes is known to be almost first‐order accurate (up to a logarithmic factor) in the energy norm induced by the bilinear form of the weak formulation, uniformly in the perturbation parameter. In the present paper the method is shown to be almost second‐order superconvergent in this energy norm for the difference between the FEM solution and the bilinear interpolant of the exact solution. This supercloseness property is used to improve the accuracy to almost second order by means of a postprocessing procedure. Numerical experiments confirm these results. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

The element‐free Galerkin method for a quasistatic contact problem with the Tresca friction in elastic materials

This paper is proposed for the error estimates of the element‐free Galerkin method for a quasistatic contact problem with the Tresca friction. The penalty method is used to impose the clamped boundary conditions. The duality algorithm is also given to deal with the non‐differentiable term in the quasistatic contact problem with the Tresca friction. The error estimates indicate that the convergence order is dependent on the nodal spacing, the time step, the largest degree of basis functions in the moving least‐squares approximation, and the penalty factor. Numerical examples demonstrate the effectiveness of the element‐free Galerkin method and verify the theoretical analysis.     

Nonintrusive reduced‐order modeling of parametrized time‐dependent partial differential equations

We propose a nonintrusive reduced‐order modeling method based on the notion of space‐time‐parameter proper orthogonal decomposition (POD) for approximating the solution of nonlinear parametrized time‐dependent partial differential equations. A two‐level POD method is introduced for constructing spatial and temporal basis functions with special properties such that the reduced‐order model satisfies the boundary and initial conditions by construction. A radial basis function approximation method is used to estimate the undetermined coefficients in the reduced‐order model without resorting to Galerkin projection. This nonintrusive approach enables the application of our approach to general problems with complicated nonlinearity terms. Numerical studies are presented for the parametrized Burgers' equation and a parametrized convection‐reaction‐diffusion problem. We demonstrate that our approach leads to reduced‐order models that accurately capture the behavior of the field variables as a function of the spatial coordinates, the parameter vector and time. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013     

Discontinuous Galerkin and multiscale variational schemes for a coupled damped nonlinear system of Schrödinger equations

In this article, we study a streamline diffusion‐based discontinuous Galerkin approximation for the numerical solution of a coupled nonlinear system of Schrödinger equations and extend the resulting method to a multiscale variational scheme. We prove stability estimates and derive optimal convergence rates due to the maximal available regularity of the exact solution. In the weak formulation, to make the underlying bilinear form coercive, it was necessary to supply the equation system with an artificial viscosity term with a small coefficient of order proportional to a power of mesh size. We justify the theory by implementing an example of an application of the time‐dependent Schrödinger equation in the coupled ultrafast laser. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

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     

Using empirical Eigenfunctions and Galerkin method to two‐phase transport models

In this article, we consider a nonlinear partial differential system describing two‐phase transports and try to recover the source term and the nonlinear diffusion term when the state variable is known at different profile times. To this end, we use a POD‐Galerkin procedure in which the proper orthogonal decomposition technique is applied to the ensemble of solutions to derive empirical eigenfunctions. These empirical eigenfunctions are then used as basis functions within a Galerkin method to transform the partial differential equation into a set of ordinary differential equations. Finally, the validation of the used method has been evaluated by some numerical examples. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 23: 456–474, 2007     

Global existence,asymptotic behavior,and uniform attractor for a nonautonomous equation

In this paper, we establish the global existence, asymptotic behavior, and uniform attractor for a nonautonomous viscoelastic equation with a delay term. Under some suitable assumptions, we firstly prove the global well‐posedness of the problem by using the Faedo–Galerkin approximations together with some energy estimates and then obtain the general decay results of the energy via suitable Lyapunov functionals. Finally, we prove the existence of uniform attractors. Copyright © 2013 John Wiley & Sons, Ltd.     

Convergence phenomena of \documentclass{article} \usepackage{amsmath,amsfonts}\pagestyle{empty}\begin{document} $\mathcal{Q}_p$ \end{document}‐elements for convection–diffusion problems

We present a numerical study for singularly perturbed convection–diffusion problems using higher order Galerkin and streamline diffusion finite element method. We are especially interested in convergence and superconvergence properties with respect to different interpolation operators. For this we investigate pointwise interpolation and vertex‐edge‐cell interpolation. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Numerical solutions of the coupled Burgers’ equation by the Galerkin quadratic B‐spline finite element method

In this paper, a coupled Burgers’ equation has been numerically solved by a Galerkin quadratic B‐spline FEM. The performance of the method has been examined on three test problems. Results obtained by the method have been compared with known exact solution and other numerical results in the literature. A Fourier stability analysis of the method is also investigated. Copyright © 2013 John Wiley & Sons, Ltd.     

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     

Turbulent flow computations on 3D unstructured grids

An edge-based finite element method is presented for the simulation of compressible turbulent flows on unstructured tetrahedral grids. A two equation k–ω turbulence model is employed and the standard Galerkin approach is used for spatial discretisation. Stabilisation of the resulting procedure is achieved by the addition of an appropriate diffusion. An explicit multistage time-stepping scheme is used to advance the solution in time to steady state. The performance of the algorithm is demonstrated for the simulation of a high Reynolds number transonic separated flow over a wing.     

Higher‐order discontinuous Galerkin method for pyramidal elements using orthogonal bases

We study finite elements of arbitrarily high‐order defined on pyramids for discontinuous Galerkin methods. We propose a new family of high‐order pyramidal finite elements using orthogonal basis functions which can be used in hybrid meshes including hexahedra, tetrahedra, wedges, and pyramids. We perform a comparison between these orthogonal functions and nodal functions for affine and non‐affine elements. Different strategies for the inversion of the mass matrix are also considered and discussed. Numerical experiments are conducted for the three dimensional Maxwell's equations. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

The convergence of fully discrete Galerkin approximations for the Benjamin–Bona–Mahony (BBM) equation

In the present paper, we analyze a second-order in time fully discrete finite element method for the BBM equation. The discretization in space is based on the standard Galerkin method, for the time discretization the Crank–Nicolson scheme is used. We also prove the convergence of a linearized Galerkin modification scheme.     

A uniformly optimal‐order error estimate for a bilinear finite element method for degenerate convection‐diffusion equations

We prove an optimal‐order error estimate in a degenerate‐diffusion weighted energy norm for bilinear Galerkin finite element methods for two‐dimensional time‐dependent convection‐diffusion equations with degenerate diffusion. In the estimate, the generic constants depend only on certain Sobolev norms of the true solution but not the lower bound of the diffusion. This estimate, combined with a known stability estimate of the true solution of the governing partial differential equations, yields an optimal‐order estimate of the Galerkin finite element method, in which the generic constants depend only on the Sobolev norms of the initial and right side data. Preliminary numerical experiments were conducted to verify these estimates numerically. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011