Superconvergence of Finite Element Approximations for the Fractional Diffusion-Wave Equation

In this paper, the error estimates of fully discrete finite element approximation for the time fractional diffusion-wave equation are discussed. Based on the standard Galerkin finite element method approach for the spatial discretization and the L1 formula for the approximation of the time fractional derivative, the fully discrete scheme for solving the constant coefficient fractional diffusion-wave equation is obtained and the superconvergence estimate is proposed and analyzed. Further, a fully discrete finite element scheme is presented for solving the variable coefficient fractional diffusion-wave equation and the corresponding error estimates are also established. Finally, numerical experiments are included to support the theoretical results.     

Superconvergence of least-squares methods for a coupled system of elliptic equations

We present a numerical method for solving a coupled system of elliptic partial differential equations (PDEs). Our method is based on the least-squares (LS) approach. We develop ellipticity estimates and error bounds for the method. The main idea of the error estimates is the establishment of supercloseness of the LS solutions, and solutions of the mixed finite element methods and Ritz projections. Using the supercloseness property, we obtain $L_2$-norm error estimates, and the error estimates for each quantity of interest show different convergence behaviors depending on the choice of the approximation spaces. Moreover, we present maximum norm error estimates and construct asymptotically exact a posteriori error estimators under mild conditions. Application to optimal control problems is briefly considered.     

Analysis of multiscale mortar mixed approximation of nonlinear elliptic equations

A multiscale mortar mixed finite element method is established to approximate non-linear second order elliptic equations. The method is based on non-overlapping domain decomposition and mortar finite element methods. The existence and uniqueness of the approximation are demonstrated, and a priori $L^2$-error estimates for the velocity and pressure are derived. An error bound for mortar pressure is proved. Convergence estimates of the mortar pressure are based on a linear interface formulation having the discrete-pressure dependent coefficient. Optimal order convergence is achieved on the fine scale by a proper choice of mortar space and polynomial degree of approximation. The quadratic convergence of the Newton–Raphson method is proved for the nonlinear algebraic system arising from the mortar mixed formulation of the problem. Numerical experiments are performed to support theoretic results.     

Analysis of the Finite Element Method for the Laplace–Beltrami Equation on Surfaces with Regions of High Curvature Using Graded Meshes

We derive error estimates for the piecewise linear finite element approximation of the Laplace–Beltrami operator on a bounded, orientable, \(C^3\), surface without boundary on general shape regular meshes. As an application, we consider a problem where the domain is split into two regions: one which has relatively high curvature and one that has low curvature. Using a graded mesh we prove error estimates that do not depend on the curvature on the high curvature region. Numerical experiments are provided.     

Solving obstacle problems with guaranteed accuracy

In this paper, we consider a numerical technique which enables us to verify the existence of solutions for some simple obstacle problems. Using the finite element approximation and constructive error estimates, we construct, on a computer, a set of solutions which satisfies the hypothesis of the Schauder fixed-point theorem for a compact map on a certain Sobolev space. We describe the numerical verification algorithm for solving a two-dimensional obstacle problems and report some numerical results.     

Parallel black box \mathcal {H}-LU preconditioning for elliptic boundary value problems

Hierarchical ( $\mathcal {H}$ -) matrices provide a data-sparse way to approximate fully populated matrices. The two basic steps in the construction of an $\mathcal {H}$ -matrix are (a) the hierarchical construction of a matrix block partition, and (b) the blockwise approximation of matrix data by low rank matrices. In the context of finite element discretisations of elliptic boundary value problems, $\mathcal {H}$ -matrices can be used for the construction of preconditioners such as approximate $\mathcal {H}$ -LU factors. In this paper, we develop a new black box approach to construct the necessary partition. This new approach is based on the matrix graph of the sparse stiffness matrix and no longer requires geometric data associated with the indices like the standard clustering algorithms. The black box clustering and a subsequent $\mathcal {H}$ -LU factorisation have been implemented in parallel, and we provide numerical results in which the resulting black box $\mathcal {H}$ -LU factorisation is used as a preconditioner in the iterative solution of the discrete (three-dimensional) convection-diffusion equation.     

$$\mathcal $$-FAINV: hierarchically factored approximate inverse preconditioners

Given a sparse matrix, its LU-factors, inverse and inverse factors typically suffer from substantial fill-in, leading to non-optimal complexities in their computation as well as their storage. In the past, several computationally efficient methods have been developed to compute approximations to these otherwise rather dense matrices. Many of these approaches are based on approximations through sparse matrices, leading to well-known ILU, sparse approximate inverse or factored sparse approximate inverse techniques and their variants. A different approximation approach is based on blockwise low rank approximations and is realized, for example, through hierarchical (\(\mathcal H\)-) matrices. While \(\mathcal H\)-inverses and \(\mathcal H\)-LU factors have been discussed in the literature, this paper will consider the construction of an approximation of the factored inverse through \(\mathcal H\)-matrices (\(\mathcal H\)-FAINV). We will describe a blockwise approach that permits to replace (exact) matrix arithmetic through approximate efficient \(\mathcal H\)-arithmetic. We conclude with numerical results in which we use approximate factored inverses as preconditioners in the iterative solution of the discretized convection–diffusion problem.     

Error Estimates of the Finite Element Method for Interior Transmission Problems

The interior transmission problem (ITP) plays an important role in the investigation of the inverse scattering problem. In this paper we propose the finite element method for solving the ITP. Based on the $\mathbb T $ -coercivity, we derive both priori error estimate and a posteriori error estimate of the finite element approximation. Numerical experiments are also included to illustrate the accuracy of the finite element method.     

Unconditionally Optimal Error Estimates of a Linearized Galerkin Method for Nonlinear Time Fractional Reaction–Subdiffusion Equations

This paper is concerned with unconditionally optimal error estimates of linearized Galerkin finite element methods to numerically solve some multi-dimensional fractional reaction–subdiffusion equations, while the classical analysis for numerical approximation of multi-dimensional nonlinear parabolic problems usually require a restriction on the time-step, which is dependent on the spatial grid size. To obtain the unconditionally optimal error estimates, the key point is to obtain the boundedness of numerical solutions in the \(L^\infty \)-norm. For this, we introduce a time-discrete elliptic equation, construct an energy function for the nonlocal problem, and handle the error summation properly. Compared with integer-order nonlinear problems, the nonlocal convolution in the time fractional derivative causes much difficulties in developing and analyzing numerical schemes. Numerical examples are given to validate our theoretical results.     

Degenerate Two-Phase Incompressible Flow IV: Local Refinement and Domain Decomposition

This is the fourth paper of a series in which we analyze mathematical properties and develop numerical methods for a degenerate elliptic-parabolic partial differential system which describes the flow of two incompressible, immiscible fluids in porous media. In this paper we describe a finite element approximation for this system on locally refined grids. This adaptive approximation is based on a mixed finite element method for the elliptic pressure equation and a Galerkin finite element method for the degenerate parabolic saturation equation. Both discrete stability and sharp a priori error estimates are established for this approximation. Iterative techniques of domain decomposition type for solving it are discussed, and numerical results are presented.     

Discontinuous Finite Volume Element Method for a Coupled Non-stationary Stokes–Darcy Problem

In this paper, a discontinuous finite volume element method was presented to solve the nonstationary Stokes–Darcy problem for the coupling fluid flow in conduits with porous media flow. The proposed numerical method is constructed on a baseline finite element family of discontinuous linear elements for the approximation of the velocity and hydraulic head, whereas the pressure is approximated by piecewise constant elements. The unique solvability of the approximate solution for the discrete problem is derived. Optimal error estimates of the semi-discretization and full discretization with backward Euler scheme in standard \(L^2\)-norm and broken \(H^1\)-norm are obtained for three discontinuous finite volume element methods (symmetric, non-symmetric and incomplete types). A series of numerical experiments are provided to illustrate the features of the proposed method, such as the optimal accuracy orders, mass conservation, capability to deal with complicated geometries, and applicability to the problems with realistic parameters.     

Anisotropic Nonconforming { EQ}_1^{rot} Quadrilateral Finite Element Approximation to Second Order Elliptic Problems

The main aim of this paper is to study the nonconforming $EQ_1^{rot}$ quadrilateral finite element approximation to second order elliptic problems on anisotropic meshes. The optimal order error estimates in broken energy norm and $L^2$ -norm are obtained, and three numerical experiments are carried out to confirm the theoretical results.     

A stabilized finite volume method for the evolutionary Stokes–Darcy system

In this paper, we propose a stabilized fully discrete finite volume method based on two local Gauss integrals for a non-stationary Stokes–Darcy problem. This stabilized method is free of stabilized parameters and uses the lowest equal-order finite element triples $P1\text{–}P1\text{–}P1$ for approximating the velocity, pressure and hydraulic head of the Stokes–Darcy model. Under a modest time step restriction in relation to physical parameters, we give the stability analysis and the error estimates for the stabilized finite volume scheme by means of a relationship between finite volume and finite element approximations with the lower order elements. Finally, a series of numerical experiments are provided to demonstrate the validity of the theoretical results.     

Inhomogeneous Dirichlet boundary condition in the a posteriori error control of the obstacle problem

We study a posteriori error control of finite element approximation of the elliptic obstacle problem with nonhomogeneous Dirichlet boundary condition. The results in the article are two fold. Firstly, we address the influence of the inhomogeneous Dirichlet boundary condition in residual based a posteriori error control of the elliptic obstacle problem. Secondly by rewriting the obstacle problem in an equivalent form, we derive a posteriori error bounds which are in simpler form and efficient. To accomplish this, we construct and use a post-processed solution ${\stackrel{?}{u}}_h$ of the discrete solution $u_h$ which satisfies the exact boundary conditions sharply although the discrete solution $u_h$ may not satisfy. We propose two post processing methods and analyze them, namely the harmonic extension and a linear extension. The theoretical results are illustrated by the numerical results.     

Unconditional convergence and optimal L2 error estimates of the Crank–Nicolson extrapolation FEM for the nonstationary Navier–Stokes equations

In this paper, we study stability and convergence of fully discrete finite element method on large timestep which used Crank–Nicolson extrapolation scheme for the nonstationary Navier–Stokes equations. This approach bases on a finite element approximation for the space discretization and the Crank–Nicolson extrapolation scheme for the time discretization. It reduces nonlinear equations to linear equations, thus can greatly increase the computational efficiency. We prove that this method is unconditionally stable and unconditionally convergent. Moreover, taking the negative norm technique, we derive the $L^2$, $H^1$-unconditionally optimal error estimates for the velocity, and the $L^2$-unconditionally optimal error estimate for the pressure. Also, numerical simulations on unconditional$L^2$-stability and convergent rates of this method are shown.     

Efficient controller synthesis for a fragment of \hbox {MTL}_{0, \infty }

In this paper we offer an efficient controller synthesis algorithm for assume-guarantee specifications of the form $\varphi _1 \wedge \varphi _2 \wedge \cdots \wedge \varphi _n \rightarrow \psi _1 \wedge \psi _2 \wedge \cdots \wedge \psi _m$ . Here, $\{\varphi _i,\psi _j\}$ are all safety-MTL $_{0, \infty }$ properties, where the sub-formulas $\{\varphi _i\}$ are supposed to specify assumptions of the environment and the sub-formulas $\{\psi _j\}$ are specifying requirements to be guaranteed by the controller. Our synthesis method exploits the engine of Uppaal-Tiga and the novel translation of safety- and co-safety-MTL $_{0, \infty }$ properties into under-approximating, deterministic timed automata. Our approach avoids determinization of Büchi automata, which is the main obstacle for the practical applicability of controller synthesis for linear-time specifications. The experiments demonstrate that the chosen specification formalism is expressive enough to specify complex behaviors. The proposed approach is sound but not complete. However, it successfully produced solutions for all the experiments. Additionally we compared our tool with Acacia+ and Unbeast, state-of-the-art LTL synthesis tools; and our tool demonstrated better timing results, when we applied both tools to the analogous specifications.     

A Posteriori Error Estimators for a Class of Variational Inequalities

In this paper, we present an a posteriori error analysis for the finite element approximation of a variational inequality. We derive a posteriori error estimators of residual type, which are shown to provide upper bounds on the discretization error for a class of variational inequalities provided the solutions are sufficiently regular. Furthermore we derive sharp a posteriori error estimators with both lower and upper error bounds for a subclass of the obstacle problem which are frequently met in many physical models. For sufficiently regular solutions, these estimates are shown to be equivalent to the discretization error in an energy type norm. Our numerical tests show that these sharp error estimators are both reliable and efficient in guiding mesh adaptivity for computing the free boundaries.     

CBS Constants & Their Role in Error Estimation for Stochastic Galerkin Finite Element Methods

Stochastic Galerkin finite element methods (SGFEMs) are commonly used to approximate solutions to PDEs with random inputs. However, the study of a posteriori error estimation strategies to drive adaptive enrichment of the associated tensor product spaces is still under development. In this work, we revisit an a posteriori error estimator introduced in Bespalov and Silvester (SIAM J Sci Comput 38(4):A2118–A2140, 2016) for SGFEM approximations of the parametric reformulation of the stochastic diffusion problem. A key issue is that the bound relating the true error to the estimated error involves a CBS (Cauchy–Buniakowskii–Schwarz) constant. If the approximation spaces associated with the parameter domain are orthogonal in a weighted \(L^2\) sense, then this CBS constant only depends on a pair of finite element spaces \(H_{1}, H_{2}\) associated with the spatial domain and their compatibility with respect to an inner product associated with a parameter-free problem. For fixed choices of \(H_{1}\), we investigate non-standard choices of \(H_{2}\) and the associated CBS constants, with the aim of designing efficient error estimators with effectivity indices close to one. When \(H_1\) and \(H_2\) satisfy certain conditions, we also prove new theoretical estimates for the CBS constant using linear algebra arguments.     

A Fictitious Domain Method with Distributed Lagrange Multiplier for Parabolic Problems With Moving Interfaces

In this paper, we study the fictitious domain method with distributed Lagrange multiplier for the jump-coefficient parabolic problems with moving interfaces. The equivalence between the fictitious domain weak form and the standard weak form of a parabolic interface problem is proved, and the uniform well-posedness of the full discretization of fictitious domain finite element method with distributed Lagrange multiplier is demonstrated. We further analyze the convergence properties for the fully discrete finite element approximation in the norms of \(L^2\), \(H^1\) and a new energy norm. On the other hand, we introduce a subgrid integration technique in order to allow the fictitious domain finite element method to be performed on the triangular meshes without doing any interpolation between the authentic domain and the fictitious domain. Numerical experiments confirm the theoretical results, and show the good performances of the proposed schemes.