Analysis of the HDG method for the stokes–darcy coupling

In this article, we introduce and analyze a hybridizable discontinuous Galerkin (HDG) method for numerically solving the coupling of fluid flow with porous media flow. Flows are governed by the Stokes and Darcy equations, respectively, and the corresponding transmission conditions are given by mass conservation, balance of normal forces, and the Beavers‐Joseph‐Saffman law. We consider a fully‐mixed formulation in which the main unknowns in the fluid are given by the stress, the vorticity, the velocity, and the trace of the velocity, whereas the velocity, the pressure, and the trace of the pressure are the unknowns in the porous medium. In addition, a suitable enrichment of the finite dimensional subspace for the stress yields optimally convergent approximations for all unknowns, as well as a superconvergent approximation of the trace variables. To do that, similarly as in previous articles dealing with development of the a priori error estimates, we use the projection‐based error analysis to simplify the corresponding study. Finally, we provide several numerical results illustrating the good performance of the proposed scheme and confirming the optimal order of convergence provided by the HDG approximation. © 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 885–917, 2017     

Analysis of DG approximations for Stokes problem based on velocity‐pseudostress formulation

In this article, we first discuss the well posedness of a modified LDG scheme of Stokes problem, considering a velocity‐pseudostress formulation. The difficulty here relies on the fact that the application of classical Babu?ka‐Brezzi theory is not easy, so we proceed in a nonstandard way. For uniqueness, we apply a discrete version of Fredholm's alternative theorem, while the a priori error analysis is done introducing suitable projections of exact solution. As a result, we prove that the method is convergent, and under suitable regularity assumptions on the exact solution, the optimal rate of convergence is guaranteed. Next, we explore two stabilizations to the previous scheme, by adding least squares type terms. For these cases, well posedness and the a priori error estimates are proved by the application of standard theory. We end this work with some numerical experiments considering our third scheme, whose results are in agreement with the theoretical properties we deduce.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1540–1564, 2017     

Numerical Analysis of a Nonlinear Singularly Perturbed Delay Volterra Integro-Differential Equation on an Adaptive Grid

In this paper, we study a nonlinear first-order singularly perturbed Volterra integro-differential equation with delay. This equation is discretized by the backward Euler for differential part and the composite numerical quadrature formula for integral part for which both an a priori and an a posteriori error analysis in the maximum norm are derived. Based on the a priori error bound and mesh equidistribution principle, we prove that there exists a mesh gives optimal first order convergence which is robust with respect to the perturbation parameter. The a posteriori error bound is used to choose a suitable monitor function and design a corresponding adaptive grid generation algorithm. Furthermore, we extend our presented adaptive grid algorithm to a class of second-order nonlinear singularly perturbed delay differential equations. Numerical results are provided to demonstrate the effectiveness of our presented monitor function. Meanwhile, it is shown that the standard arc-length monitor function is unsuitable for this type of singularly perturbed delay differential equations with a turning point.     

A mixed‐FEM formulation for nonlinear incompressible elasticity in the plane

This article deals with an expanded mixed finite element formulation, based on the Hu‐Washizu principle, for a nonlinear incompressible material in the plane. We follow our related previous works and introduce both the stress and the strain tensors as further unknowns, which yields a two‐fold saddle point operator equation as the corresponding variational formulation. A slight generalization of the classical Babu?ka‐Brezzi's theory is applied to prove unique solvability of the continuous and discrete formulations, and to derive the corresponding a priori error analysis. An extension of the well‐known PEERS space is used to define an stable associated Galerkin scheme. Finally, we provide an a posteriori error analysis based on the classical Bank‐Weiser approach. © 2002 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 18: 105–128, 2002     

Multiscale finite element discretizations based on local defect correction for the biharmonic eigenvalue problem of plate buckling

In this paper, we study the multiscale finite element discretizations about the biharmonic eigenvalue problem of plate buckling. On the basis of the work of Dai and Zhou (SIAM J. Numer. Anal. 46[1] [2008] 295‐324), we establish a three‐scale scheme, a multiscale discretization scheme, and the associated parallel version based on local defect correction. We first prove a local priori error estimate of finite element approximations, then give the error estimates of multiscale discretization schemes. Theoretical analysis and numerical experiments indicate that our schemes are suitable and efficient for eigenfunctions with local low smoothness.     

A space‐time spectral method for one‐dimensional time fractional convection diffusion equations

In this paper, we propose a space‐time spectral method for solving a class of time fractional convection diffusion equations. Because both fractional derivative and spectral method have global characteristics in bounded domains, we propose a space‐time spectral‐Galerkin method. The convergence result of the method is proved by providing a priori error estimate. Numerical results further confirm the expected convergence rate and illustrate the versatility of our method. Copyright © 2016 John Wiley & Sons, Ltd.     

Guaranteed error bounds for finite element approximations of noncoercive elliptic problems and their applications

In this paper, we discuss with guaranteed a priori and a posteriori error estimates of finite element approximations for not necessarily coercive linear second order Dirichlet problems. Here, ‘guaranteed’ means we can get the error bounds in which all constants included are explicitly given or represented as a numerically computable form. Using the invertibility condition of concerning elliptic operator, guaranteed a priori and a posteriori error estimates are formulated. This kind of estimates plays essential and important roles in the numerical verification of solutions for nonlinear elliptic problems. Several numerical examples that confirm the actual effectiveness of the method are presented.     

Discontinuous Least-Squares finite element method for the Div-Curl problem

In this paper, we consider the div-curl problem posed on nonconvex polyhedral domains. We propose a least-squares method based on discontinuous elements with normal and tangential continuity across interior faces, as well as boundary conditions, weakly enforced through a properly designed least-squares functional. Discontinuous elements make it possible to take advantage of regularity of given data (divergence and curl of the solution) and obtain convergence also on nonconvex domains. In general, this is not possible in the least-squares method with standard continuous elements. We show that our method is stable, derive a priori error estimates, and present numerical examples illustrating the method.     

A local radial basis function collocation method to solve the variable‐order time fractional diffusion equation in a two‐dimensional irregular domain

The local radial basis function (RBF) method is a promising solver for variable‐order time fractional diffusion equation (TFDE), as it overcomes the computational burden of the traditional global method. Application of the local RBF method is limited to Fickian diffusion, while real‐world diffusion is usually non‐Fickian in multiple dimensions. This article is the first to extend the application of the local RBF method to two‐dimensional, variable‐order, time fractional diffusion equation in complex shaped domains. One of the main advantages of the local RBF method is that only the nodes located in the subdomain, surrounding the local point, need to be considered when calculating the numerical solution at this point. This approach can perform well with large scale problems and can also mitigate otherwise ill‐conditioned problems. The proposed numerical approach is checked against two examples with curved boundaries and known analytical solutions. Shape parameter and subdomain node number are investigated for their influence on the accuracy of the local RBF solution. Furthermore, quantitative analysis, based on root‐mean‐square error, maximum absolute error, and maximum error of the partial derivative indicates that the local RBF method is accurate and effective in approximating the variable‐order TFDE in two‐dimensional irregular domains.     

Convergence analysis of a regularized approximation for solving Fredholm integral equations of the first kind

In this paper, we suggest a convergence analysis for solving Fredholm integral equations of the first kind using Tikhonov regularization under supremum norm. We also provide an a priori parameter choice strategy for choosing the regularization parameter and obtain an error estimate.     

Finite element theory on curved domains with applications to discontinuous Galerkin finite element methods

In this paper we provide key estimates used in the stability and error analysis of discontinuous Galerkin finite element methods (DGFEMs) on domains with curved boundaries. In particular, we review trace estimates, inverse estimates, discrete Poincaré–Friedrichs' inequalities, and optimal interpolation estimates in noninteger Hilbert–Sobolev norms, that are well known in the case of polytopal domains. We also prove curvature bounds for curved simplices, which does not seem to be present in the existing literature, even in the polytopal setting, since polytopal domains have piecewise zero curvature. We demonstrate the value of these estimates, by analyzing the IPDG method for the Poisson problem, introduced by Douglas and Dupont, and by analyzing a variant of the hp-DGFEM for the biharmonic problem introduced by Mozolevski and Süli. In both cases we prove stability estimates and optimal a priori error estimates. Numerical results are provided, validating the proven error estimates.     

On an initial‐boundary value problem for a wide‐angle parabolic equation in a waveguide with a variable bottom

We consider the third‐order Claerbout‐type wide‐angle parabolic equation (PE) of underwater acoustics in a cylindrically symmetric medium consisting of water over a soft bottom B of range‐dependent topography. There is strong indication that the initial‐boundary value problem for this equation with just a homogeneous Dirichlet boundary condition posed on B may not be well‐posed, for example when B is downsloping. We impose, in addition to the above, another homogeneous, second‐order boundary condition, derived by assuming that the standard (narrow‐angle) PE holds on B, and establish a priori H² estimates for the solution of the resulting initial‐boundary value problem for any bottom topography. After a change of the depth variable that makes B horizontal, we discretize the transformed problem by a second‐order accurate finite difference scheme and show, in the case of upsloping and downsloping wedge‐type domains, that the new model gives stable and accurate results. We also present an alternative set of boundary conditions that make the problem exactly energy conserving; one of these conditions may be viewed as a generalization of the Abrahamsson–Kreiss boundary condition in the wide‐angle case. Copyright © 2008 John Wiley & Sons, Ltd.     

Identifying an unknown source in the Poisson equation by the method of Tikhonov regularization in Hilbert scales

In this paper, we consider the problem for identifying the unknown source in the Poisson equation. The Tikhonov regularization method in Hilbert scales is extended to deal with illposedness of the problem and error estimates are obtained with an a priori strategy and an a posteriori choice rule to find the regularization parameter. The user does not need to estimate the smoothness parameter and the a priori bound of the exact solution when the a posteriori choice rule is used. Numerical examples show that the proposed method is effective and stable.     

Discontinuous Galerkin method for the coupled Stokes‐Biot model

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.     

On the p-version approximation in the boundary element method for a variational inequality of the second kind modelling unilateral contact and given friction

In this paper we complement recent work of Maischak and Stephan on adaptive hp-versions of the BEM for unilateral Signorini problems, respectively on FEM-BEM coupling in its h-version for a nonlinear transmission problem modelling Coulomb friction contact. Here we focus on the boundary element method in its p-version to treat a scalar variational inequality of the second kind that models unilateral contact and Coulomb friction in elasticity together. This leads to a nonconforming discretization scheme. In contrast to the work cited above and to a related paper of Guediri on a boundary variational inequality of the second kind modelling friction we take the quadrature error of the friction functional into account of the error analysis. At first without any regularity assumptions, we prove convergence of the BEM Galerkin approximation in the energy norm. Then under mild regularity assumptions, we establish an a priori error estimate that is based on a novel Céa–Falk lemma for abstract variational inequalities of the second kind.     

Discontinuous Galerkin finite element methods for variational inequalities of first and second kinds

We develop the error analysis for the h‐version of the discontinuous Galerkin finite element discretization for variational inequalities of first and second kinds. We establish an a priori error estimate for the method which is of optimal order in a mesh dependant as well as L²‐norm.© 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2007     

Finite element methods for elliptic optimal control problems with boundary observations

We study in this paper the finite element approximations to elliptic optimal control problems with boundary observations. The main feature of this kind of optimal control problems is that the observations or measurements are the outward normal derivatives of the state variable on the boundary, this reduces the regularity of solutions to the optimal control problems. We propose two kinds of finite element methods: the standard FEM and the mixed FEM, to efficiently approximate the underlying optimal control problems. For both cases we derive a priori error estimates for problems posed on polygonal domains. Some numerical experiments are carried out at the end of the paper to support our theoretical findings.     

Optimal impulse control on an unbounded domain with nonlinear cost functions

In this paper we consider the optimal impulse control of a system which evolves randomly in accordance with a homogeneous diffusion process in ℜ¹. Whenever the system is controlled a cost is incurred which has a fixed component and a component which increases with the magnitude of the control applied. In addition to these controlling costs there are holding or carrying costs which are a positive function of the state of the system. Our objective is to minimize the expected discounted value of all costs over an infinite planning horizon. Under general assumptions on the cost functions we show that the value function is a weak solution of a quasi-variational inequality and we deduce from this solution the existence of an optimal impulse policy. The computation of the value function is performed by means of the Finite Element Method on suitable truncated domains, whose convergence is discussed. Mathematics Subject Classification: 49J40, 60G40, 65N30     

Finite element method for a nonsmooth elliptic equation

In this paper, we study the finite element method for a non-smooth elliptic equation. Error analysis is presented, including a priori and a posteriori error estimates as well as superconvergence analysis. We also propose two algorithms for solving the underlying equation. Numerical experiments are employed to confirm our error estimations and the efficiency of our algorithms.     

Projection stabilized nonconforming finite element methods for the stokes problem

In this article we study a projection‐stabilized nonconforming finite element discretization of the Stokes problem. We present a priori error analysis and give a recovery‐based a posteriori error estimator for the considered problem. Numerical results illustrate the theoretical performance of the error estimator. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 218–240, 2017