A fast discontinuous finite element discretization for the space‐time fractional diffusion‐wave equation

In this article, we study fast discontinuous Galerkin finite element methods to solve a space‐time fractional diffusion‐wave equation. We introduce a piecewise‐constant discontinuous finite element method for solving this problem and derive optimal error estimates. Importantly, a fast solution technique to accelerate Toeplitz matrix‐vector multiplications which arise from discontinuous Galerkin finite element discretization is developed. This fast solution technique is based on fast Fourier transform and it depends on the special structure of coefficient matrices. In each temporal step, it helps to reduce the computational work from required by the traditional methods to log , where is the size of the coefficient matrices (number of spatial grid points). Moreover, the applicability and accuracy of the method are verified by numerical experiments including both continuous and discontinuous examples to support our theoretical analysis.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 2043–2061, 2017     

The fictitious domain method with L2‐penalty for the Stokes problem with the Dirichlet boundary condition

We consider the fictitious domain method with L²‐penalty for the Stokes problem with the Dirichlet boundary condition. First, we investigate the error estimates for the penalty method at the continuous level. We obtain the convergence of order in H¹‐norm for the velocity and in L²‐norm for the pressure, where is the penalty parameter. The L²‐norm error estimate for the velocity is upgraded to . Moreover, we derive the a priori estimates depending on for the solution of the penalty problem. Next, we apply the finite element approximation to the penalty problem using the P1/P1 element with stabilization. For the discrete penalty problem, we prove the error estimate in H¹‐norm for the velocity and in L²‐norm for the pressure, where h denotes the discretization parameter. For the velocity in L²‐norm, the convergence rate is improved to . The theoretical results are verified by the numerical experiments.     

New interpolation error estimates and a posteriori error analysis for linear parabolic interface problems

We derive residual‐based a posteriori error estimates of finite element method for linear parabolic interface problems in a two‐dimensional convex polygonal domain. Both spatially discrete and fully discrete approximations are analyzed. While the space discretization uses finite element spaces that are allowed to change in time, the time discretization is based on the backward Euler approximation. The main ingredients used in deriving a posteriori estimates are new Clément type interpolation estimates and an appropriate adaptation of the elliptic reconstruction technique introduced by (Makridakis and Nochetto, SIAM J Numer Anal 4 (2003), 1585–1594). We use only an energy argument to establish a posteriori error estimates with optimal order convergence in the ‐norm and almost optimal order in the ‐norm. The interfaces are assumed to be of arbitrary shape but are smooth for our purpose. Numerical results are presented to validate our derived estimators. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 570–598, 2017     

Superconvergence of Ritz‐Galerkin finite element approximations for second order elliptic problems

In this paper, the author derives an ‐superconvergence for the piecewise linear Ritz‐Galerkin finite element approximations for the second‐order elliptic equation equipped with Dirichlet boundary conditions. This superconvergence error estimate is established between the finite element solution and the usual Lagrange nodal point interpolation of the exact solution, and thus the superconvergence at the nodal points of each element. The result is based on a condition for the finite element partition characterized by the coefficient tensor and the usual shape functions on each element, called ‐equilateral assumption in this paper. Several examples are presented for the coefficient tensor and finite element triangulations which satisfy the conditions necessary for superconvergence. Some numerical experiments are conducted to confirm this new theory of superconvergence.     

Least‐squares method for the Oseen equation

This article studies the least‐squares finite element method for the linearized, stationary Navier–Stokes equation based on the stress‐velocity‐pressure formulation in d dimensions (d = 2 or 3). The least‐squares functional is simply defined as the sum of the squares of the L² norm of the residuals. It is shown that the homogeneous least‐squares functional is elliptic and continuous in the norm. This immediately implies that the a priori error estimate of the conforming least‐squares finite element approximation is optimal in the energy norm. The L² norm error estimate for the velocity is also established through a refined duality argument. Moreover, when the right‐hand side f belongs only to , we derive an a priori error bound in a weaker norm, that is, the norm. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1289–1303, 2016     

Convergence analysis of a new mixed finite element method for Biot's consolidation model

In this article, we propose a mixed finite element method for the two‐dimensional Biot's consolidation model of poroelasticity. The new mixed formulation presented herein uses the total stress tensor and fluid flux as primary unknown variables as well as the displacement and pore pressure. This method is based on coupling two mixed finite element methods for each subproblem: the standard mixed finite element method for the flow subproblem and the Hellinger–Reissner formulation for the mechanical subproblem. Optimal a‐priori error estimates are proved for both semidiscrete and fully discrete problems when the Raviart–Thomas space for the flow problem and the Arnold–Winther space for the elasticity problem are used. In particular, optimality in the stress, displacement, and pressure has been proved in when the constrained‐specific storage coefficient is strictly positive and in the weaker norm when is nonnegative. We also present some of our numerical results.Copyright © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1189–1210, 2014     

A priori and a posteriori error analysis for the mixed discontinuous Galerkin finite element approximations of the biharmonic problems

In this article, a new mixed discontinuous Galerkin finite element method is proposed for the biharmonic equation in two or three‐dimension space. It is amenable to an efficient implementation displaying new convergence properties. Through an auxiliary variable , we rewrite the problem into a two‐order system. Then, the a priori error estimates are derived in L² norm and in the broken DG norm for both u and p. We prove that, when polynomials of degree r () are used, we obtain the optimal convergence rate of order r + 1 in L² norm and of order r in DG norm for u, and the order r in both norms for . The numerical experiments illustrate the theoretic order of convergence. For the purpose of adaptive finite element method, the a posteriori error estimators are also proposed and proved to field a sharp upper bound. We also provide numerical evidence that the error estimators and indicators can effectively drive the adaptive strategies. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 318–353, 2017     

Error analysis of mixed finite element method for Poisson‐Nernst‐Planck system

To improve the convergence rate in L² norm from suboptimal to optimal for both electrostatic potential and ionic concentrations in Poisson‐Nernst‐Planck (PNP) system, we propose the mixed finite element method in this article to discretize the electrostatic potential equation, and still use the standard finite element method to discretize the time‐dependent ionic concentrations equations. Optimal error estimates in norm for the electrostatic potential, and in and norms for the ionic concentrations are attained. As a by‐product, the electric field can also achieve a higher approximation order in contrast with the standard finite element method for PNP system. Numerical experiments are performed to validate the theoretical results.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1924–1948, 2017     

Unconditional superconvergence analysis of an H1‐galerkin mixed finite element method for nonlinear Sobolev equations

An efficient H¹‐Galerkin mixed finite element method (MFEM) is presented with and zero order Raviart‐Thomas elements for the nonlinear Sobolev equations. On one hand, the existence and uniqueness of the solutions of the semidiscrete approximation scheme are proved and the super close results of order for the original variable u in a broken H¹ norm and the auxiliary variable in norm are deduced without the boundedness of the numerical solution in ‐norm. Conversely, a linearized Crank‐Nicolson fully discrete scheme with the unconditional super close property is also developed through a new approach, while previous literature always require certain time step conditions (see the references below). Finally, a numerical experiment is included to illustrate the feasibility of the proposed method. Here h is the subdivision parameter and τ is the time step.     

Three‐dimensional quadratic nonconforming brick element

A new nonconforming brick element with quadratic convergence for the energy norm is introduced. The nonconforming element consists of on a cube [?1,1]³, and 14 degree of freedom (DOF). Two types of DOF are introduced. One consists of the value at the eight vertices and six face‐centroids and the other consists of the value at the eight vertices and the integration value of six faces. Error estimates of optimal order are derived in both broken energy and norms for second‐order elliptic problems. If a genuine hexahedron, which is not a parallelepiped, is included in the partition, the proposed element is also convergent, but with a lower order. Copyright © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 158–174, 2014     

On a nonoverlapping additive Schwarz method for h‐p discontinuous Galerkin discretization of elliptic problems

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     

Weak Galerkin finite element methods for linear parabolic integro‐differential equations

The semidiscrete and fully discrete weak Galerkin finite element schemes for the linear parabolic integro‐differential equations are proposed. Optimal order error estimates are established for the corresponding numerical approximations in both and norms. Numerical experiments illustrating the error behaviors are provided.© 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1357–1377, 2016     

Superconvergence of conforming finite element for fourth‐order singularly perturbed problems of reaction diffusion type in 1D

We consider conforming finite element approximation of fourth‐order singularly perturbed problems of reaction diffusion type. We prove superconvergence of standard C¹ finite element method of degree p on a modified Shishkin mesh. In particular, a superconvergence error bound of in a discrete energy norm is established. The error bound is uniformly valid with respect to the singular perturbation parameter ?. Numerical tests indicate that the error estimate is sharp. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 550–566, 2014     

Linear continuous interior penalty finite element method for Helmholtz equation With High Wave Number: One‐Dimensional Analysis

This article addresses the properties of continuous interior penalty (CIP) finite element solutions for the Helmholtz equation. The ‐version of the CIP finite element method with piecewise linear approximation is applied to a one‐dimensional (1D) model problem. We first show discrete well posedness and convergence results, using the imaginary part of the stabilization operator, for the complex Helmholtz equation. Then we consider a method with real valued penalty parameter and prove an error estimate of the discrete solution in the ‐norm, as the sum of best approximation error plus a pollution term that is the order of the phase difference. It is proved that the pollution effect can be eliminated by selecting the penalty parameter appropriately. As a result of this analysis, thorough and rigorous understanding of the error behavior throughout the range of convergence is gained. Numerical results are presented that show sharpness of the error estimates and highlight some phenomena of the discrete solution behavior. In particular, we give numerical evidence that the optimal penalty parameter obtained in the 1D case also works very well for the CIP‐FEM on two‐dimensional Cartesian grids.© 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1378–1410, 2016     

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.     

Asymptotically exact a posteriori local discontinuous Galerkin error estimates for the one‐dimensional second‐order wave equation

In this article, we analyze a residual‐based a posteriori error estimates of the spatial errors for the semidiscrete local discontinuous Galerkin (LDG) method applied to the one‐dimensional second‐order wave equation. These error estimates are computationally simple and are obtained by solving a local steady problem with no boundary condition on each element. We apply the optimal L² error estimates and the superconvergence results of Part I of this work [Baccouch, Numer Methods Partial Differential Equations 30 (2014), 862–901] to prove that, for smooth solutions, these a posteriori LDG error estimates for the solution and its spatial derivative, at a fixed time, converge to the true spatial errors in the L²‐norm under mesh refinement. The order of convergence is proved to be , when p‐degree piecewise polynomials with are used. As a consequence, we prove that the LDG method combined with the a posteriori error estimation procedure yields both accurate error estimates and superconvergent solutions. Our computational results show higher convergence rate. We further prove that the global effectivity indices, for both the solution and its derivative, in the L²‐norm converge to unity at rate while numerically they exhibit and rates, respectively. Numerical experiments are shown to validate the theoretical results. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1461–1491, 2015     

Variational multiscale proper orthogonal decomposition: Navier‐stokes equations

We develop a variational multiscale proper orthogonal decomposition (POD) reduced‐order model (ROM) for turbulent incompressible Navier‐Stokes equations. Under two assumptions on the underlying finite element approximation and the generation of the POD basis, the error analysis of the full discretization of the ROM is presented. All error contributions are considered: the spatial discretization error (due to the finite element discretization), the temporal discretization error (due to the backward Euler method), and the POD truncation error. Numerical tests for a three‐dimensional turbulent flow past a cylinder at Reynolds number show the improved physical accuracy of the new model over the standard Galerkin and mixing‐length POD ROMs. The high computational efficiency of the new model is also showcased. Finally, the theoretical error estimates are confirmed by numerical simulations of a two‐dimensional Navier‐Stokes problem. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 641–663, 2014     

Least‐squares spectral method for velocity‐vorticity‐pressure form of the Stokes equations

The aim of this article is to present and analyze first‐order system least‐squares spectral method for the Stokes equations in two‐dimensional spaces. The Stokes equations are transformed into a first‐order system of equations by introducing vorticity as a new variable. The least‐squares functional is then defined by summing up the ‐ and ‐norms of the residual equations. The ‐norm in the least‐squares functional is replaced by suitable operator. Continuous and discrete homogeneous least‐squares functionals are shown to be equivalent to ‐norm of velocity and ‐norm of vorticity and pressure for spectral Galerkin and pseudospectral method. The spectral convergence of the proposed methods are given and the theory is validated by numerical experiment. Mass conservation is also briefly investigated. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 661–680, 2016     

Local error estimates of the finite element method for an elliptic problem with a Dirac source term

The solutions of elliptic problems with a Dirac measure right‐hand side are not in dimension and therefore the convergence of the finite element solutions is suboptimal in the ‐norm. In this article, we address the numerical analysis of the finite element method for the Laplace equation with Dirac source term: we consider, in dimension 3, the Dirac measure along a curve and, in dimension 2, the punctual Dirac measure. The study of this problem is motivated by the use of the Dirac measure as a reduced model in physical problems, for which high accuracy of the finite element method at the singularity is not required. We show a quasioptimal convergence in the ‐norm, for on subdomains which exclude the singularity; in the particular case of Lagrange finite elements, an optimal convergence in ‐norm is shown on a family of quasiuniform meshes. Our results are obtained using local Nitsche and Schatz‐type error estimates, a weak version of Aubin‐Nitsche duality lemma and a discrete inf‐sup condition. These theoretical results are confirmed by numerical illustrations.     

A C0‐weak Galerkin finite element method for fourth‐order elliptic problems

This article proposes and analyzes a C⁰‐weak Galerkin (WG) finite element method for solving the biharmonic equation in two‐dimensional and three‐dimensional. The new WG method uses continuous piecewise‐polynomial approximations of degree for the unknown u and discontinuous piecewise‐polynomial approximations of degree k for the trace of on the interelement boundaries. Optimal error estimates are obtained in H², H¹, and L² norms. Numerical experiments illustrate and confirm the theoretical results. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1090–1104, 2016