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As an extension of the finite element method, the virtual element method (VEM) can handle very general polygonal meshes, making it very suitable for non-matching meshes. In (Wriggers et al. in Comput. Mech. 58:1039–1050, 2016), the lowest-order virtual element method was applied to solve the contact problem of two elastic bodies on non-matching meshes. The numerical experiments showed the robustness and accuracy of the virtual element scheme. In this paper, we establish a priori error estimate of the virtual element method for the contact problem and prove that the lowest-order VEM achieves linear convergence order, which is optimal. [ABSTRACT FROM AUTHOR]

In this paper, we express and analyze mixed discontinuous Galerkin(DG) methods of biharmonic eigenvalue problems as well as present the error analysis for them. The analysis consists of two parts. First, we propose a residual-based a posteriori error estimator in the approximate eigenfunctions and eigenvalues. The error in the eigenfunctions is measured both in the L2 and DG (energy-like) norms. In addition, we prove that if the error estimator converges to zero, then the distance of the computed eigenfunction from the true eigenspace also converges to zero, and so, the computed eigenvalue converges to a true eigenvalue. Next, we establish an a priori error estimate with the optimal convergence order both in the L2 and DG norms. We show that the methods can retain the same convergence properties they enjoy in the case of source problems. [ABSTRACT FROM AUTHOR]

We extend the formulation and a priori error analysis given by Claes Johnson (1993) from the acoustic wave equation to a Voigt and Maxwell–Zener viscodynamic system incorporating Rayleigh damping. The elastic term in the Rayleigh damping introduces a multiplicative T 1 ∕ 2 growth in the constant but otherwise the error bound is consistent with that obtained by Johnson, with a constant that grows a priori with T 1 ∕ 2 and also with norms of the solution. Gronwall's inequality is not used and so we can expect that this bound is of high enough quality to afford confidence in long-time integration. The viscoelasticity is modelled by internal variables that evolve according to ordinary differential equations and so the system shares similarities with dispersive Debye and Drude metamaterial models currently being studied in electromagnetism, as well as to acoustic metamaterial systems. This appears to be the first time an a priori error analysis has been given for DG-in-time treatment of dispersive problems of this type. • The formulation and error analysis is given for viscodynamics with Rayleigh damping. • Gronwall's inequality is not used. • Viscoelasticity is modelled by ODE's for internal variables. • The first a priori error analysis for DG-in-time dispersive problems of this type. [ABSTRACT FROM AUTHOR]

A new minimization principle for the Poisson equation using two variables – the solution and the gradient of the solution – is introduced. This principle allows us to use any conforming finite element spaces for both variables, where the finite element spaces do not need to satisfy the so-called inf–sup condition. A numerical example demonstrates the superiority of this approach. [ABSTRACT FROM AUTHOR]

In this article, we apply a modified weak Galerkin method to solve variational inequality of the first kind which includes Signorini and obstacle problems. Optimal order a priori error estimates in the energy norm are derived. We also provide some numerical experiments to validate the theoretical results.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1459-1474, 2017 [ABSTRACT FROM AUTHOR]

This article discusses a priori and a posteriori error estimates of discontinuous Galerkin finite element method for optimal control problem governed by the transport equation. We use variational discretization concept to discretize the control variable and discontinuous piecewise linear finite elements to approximate the state and costate variable. Based on the error estimates of discontinuous Galerkin finite element method for the transport equation, we get a priori and a posteriori error estimates for the transport equation optimal control problem. Finally, two numerical experiments are carried out to confirm the theoretical analysis.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1493-1512, 2017 [ABSTRACT FROM AUTHOR]

In this paper we investigate a semi-discreteH1-Galerkin mixed finite element approximation of one kind of nolocal second order nonlinear hyperbolic equation, which is often used to describe vibration of an elastic string. A priori error estimates for the semi-discrete scheme are derived. A fully discrete scheme is constructed and one numerical example is given to verify the theoretical findings. [ABSTRACT FROM PUBLISHER]

This work combines the consistency in lower-order differential operators with external approximations of functional spaces to obtain error estimates for finite difference finite volume schemes on unstructured nonuniform meshes. This combined approach is first applied to a one-dimensional elliptic boundary value problem on nonuniform meshes, and a first-order convergence rate is obtained, which agrees with the results previously reported. The approach is also applied to the staggered Marker-and-Cell scheme for the two-dimensional incompressible Stokes problem on unstructured meshes. A first-order convergence rate is obtained, which improves over a previously reported result in that it also holds on unstructured meshes. For both problems considered in this work, the convergence rate is one order higher on meshes satisfying special requirements. [ABSTRACT FROM AUTHOR]

We enrich the known results about coupled fixed and best proximity points of cyclic contraction ordered pair of maps. The uniqueness of the coupled best proximity points for cyclic contraction ordered pair of maps in a uniformly convex Banach space is proven. We find a priori and a posteriori error estimates for the coupled best proximity points, obtained by sequences of successive iterations, when the underlying Banach space has modulus of convexity of power type. A looser conditions are presented for the existence and uniqueness of coupled fixed points of a cyclic contraction ordered pair of maps in a complete metric space and a priori, a posteriori error estimates and the rate of convergence for the coupled fixed points are obtained for the sequences of successive iterations. We apply these results for solving systems of integral equations, systems of linear and nonlinear equations. [ABSTRACT FROM AUTHOR]

In the current paper, we derive the a priori error analysis ( hp version) of the streamline diffusion DG finite element approximation for optimal distributed control problem governed by the first order linear hyperbolic equation. We present the stability of such method, obtain the a priori error upper bound for the state and the control approximation, and prove the convergence of numerical method. For the optimal control problem, these estimates are apparently not available in the literature. [ABSTRACT FROM AUTHOR]

This paper investigates the augmented plane wave methods which are widely used in full-potential electronic structure calculations. These methods introduce basis functions that describe different regions using different discretization schemes. We construct a nonconforming method based on this idea and present an a priori error analysis for both linear Schrödinger type equations and nonlinear Kohn-Sham equations. Some numerical experiments are presented to support our theory. [ABSTRACT FROM AUTHOR]

In this paper, we apply the weakly over-penalized symmetric interior penalty method to solve some variational inequalities which include the Signorini problem and the obstacle problem. Optimal a priori error estimates in energy norm are derived. Some numerical tests are provided to confirm our theoretical analysis. [ABSTRACT FROM AUTHOR]

This paper constructs a local generalized finite element basis for elliptic problems with heterogeneous and highly varying coefficients. The basis functions are solutions of local problems on vertex patches. The error of the corresponding generalized finite element method decays exponentially with respect to the number of layers of elements in the patches. Hence, on a uniform mesh of size H, patches of diameter H log(1/H) are sufficient to preserve a linear rate of convergence in H without pre-asymptotic or resonance effects. The analysis does not rely on regularity of the solution or scale separation in the coefficient. This result motivates new and justifies old classes of variational multiscale methods. [ABSTRACT FROM AUTHOR]

This paper proposes and analyses two numerical methods for solving elliptic partial differential equations with random coefficients, under the finite noise assumption. First, the stochastic discontinuous Galerkin method represents the stochastic solution in a Galerkin framework. Second, the Monte Carlo discontinuous Galerkin method samples the coefficients by a Monte Carlo approach. Both methods discretize the differential operators by the class of interior penalty discontinuous Galerkin methods. Error analysis is obtained. Numerical results show the sensitivity of the expected value and variance with respect to the penalty parameter of the spatial discretization. [ABSTRACT FROM AUTHOR]

This paper reviews standard oversampling strategies as performed in the multiscale finite element method (MsFEM). Common to those approaches is that the oversampling is performed in the full space restricted to a patch including coarse finite element functions. We suggest, by contrast, performing local computations with the additional constraint that trial and test functions be linear independent from coarse finite element functions. This approach reinterprets the variational multiscale method in the context of computational homogenization. This connection gives rise to a general fully discrete error analysis for the proposed multiscale method with constrained oversampling without any resonance effects. In particular, we are able to give the first rigorous proof of convergence for an MsFEM with oversampling. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we provide a piecewise linear Galerkin approximation of a second order transmission problem across a highly conductive prefractal layer of von Koch type. We firstly generate an appropriate mesh adapted to the geometric shape of the interface and then we construct a refinement algorithm consistent with a suitable estimate in appropriate weighted Sobolev spaces. We also obtain a quasi-optimal error estimate in the energy norm and finally we demonstrate the validity of our theory through numerical tests. [Copyright &y& Elsevier]

The goal of this paper is to study a mixed finite element approximation of the general convex optimal control problems governed by quasilinear elliptic partial differential equations. The state and co-state are approximated by the lowest order Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise constant functions. We derive a priori error estimates both for the state variables and the control variable. Finally, some numerical examples are given to demonstrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper we study the finite element approximation of Dirichlet boundary control problems governed by elliptic PDEs. Based on a mixed variational scheme, we establish a mixed finite element approximation to the underlying optimal control problem. We consider the optimal control problems posed on both polygonal and general smooth domains, and we derive a priori error estimates for optimal control, state, and adjoint state. The optimal and quasi-optimal error estimates are obtained for problems on polygonal and smooth domains, respectively. Numerical experiments are provided to confirm our theoretical results. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we consider the finite element approximation of an elliptic optimal control problem. Based on an assumption on the adjoint state of the continuous problem with a small parameter, which represents a regularization of the bang–bang type control problem, we derive robust a priori error estimates for optimal control and state and a posteriori error estimate is also presented. Numerical experiments confirm our theoretical results. [Copyright &y& Elsevier]

In this paper we analyze the Local Discontinuous Galerkin (LDG) method for the constrained optimal control problem governed by the unsteady convection diffusion equations. A priori error estimates are obtained for both the state, the adjoint state and the control. For the discretization of the control we discuss two different approaches which have been used for elliptic optimal control problem. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents an a priori error analysis of the hp-version of the boundary element method for the electric field integral equation on a piecewise plane (open or closed) Lipschitz surface. We use -conforming discretisations with Raviart–Thomas elements on a sequence of quasi-uniform meshes of triangles and/or parallelograms. Assuming the regularity of the solution to the electric field integral equation in terms of Sobolev spaces of tangential vector fields, and based upon the known quasi-optimal convergence, we prove an a priori error estimate of the method in the energy norm. This estimate proves the expected rate of convergence with respect to the mesh parameter h and the polynomial degree p. [Copyright &y& Elsevier]

Abstract: This paper studies the coupling between anisotropic mesh adaptation and goal-oriented error estimate. The former is very well suited to the control of the interpolation error. It is generally interpreted as a local geometric error estimate. On the contrary, the latter is preferred when studying approximation errors for PDEs. It generally involves non local error contributions. Consequently, a full and strong coupling between both is hard to achieve due to this apparent incompatibility. This paper shows how to achieve this coupling in three steps. First, a new a priori error estimate is proved in a formal framework adapted to goal-oriented mesh adaptation for output functionals. This estimate is based on a careful analysis of the contributions of the implicit error and of the interpolation error. Second, the error estimate is applied to the set of steady compressible Euler equations which are solved by a stabilized Galerkin finite element discretization. A goal-oriented error estimation is derived. It involves the interpolation error of the Euler fluxes weighted by the gradient of the adjoint state associated with the observed functional. Third, rewritten in the continuous mesh framework, the previous estimate is minimized on the set of continuous meshes thanks to a calculus of variations. The optimal continuous mesh is then derived analytically. Thus, it can be used as a metric tensor field to drive the mesh adaptation. From a numerical point of view, this method is completely automatic, intrinsically anisotropic, and does not depend on any a priori choice of variables to perform the adaptation. 3D examples of steady flows around supersonic and transsonic jets are presented to validate the current approach and to demonstrate its efficiency. [Copyright &y& Elsevier]

Abstract: Two parallel domain decomposition procedures for solving initial-boundary value problems of parabolic partial differential equations are proposed. One is the extended D–D type algorithm, which extends the explicit/implicit conservative Galerkin domain decomposition procedures, given in , from a rectangle domain and its decomposition that consisted of a stripe of sub-rectangles into a general domain and its general decomposition with a net-like structure. An almost optimal error estimate, without the factor given in Dawson–Dupont’s error estimate, is proved. Another is the parallel domain decomposition algorithm of improved D–D type, in which an additional term is introduced to produce an approximation of an optimal error accuracy in -norm. [Copyright &y& Elsevier]

Abstract: The Ciarlet–Raviart mixed finite element approximation is constructed to solve the constrained optimal control problem governed by the first bi-harmonic equation. The optimality conditions consisting of the state and the co-state equations is derived. Also, the a priori error estimates are analyzed. In the analysis of the a priori error estimates, the improved convergent rate of the higher order than existed results is proved. Some numerical experiments are performed to confirm the theoretical analysis for the a priori error estimate. [Copyright &y& Elsevier]

Abstract: A priori error estimates in the - and -norms are established for the finite element method applied to the exterior Helmholtz problem, with modified Dirichlet-to-Neumann (MDtN) boundary condition. The error estimates include the effect of truncation of the MDtN boundary condition as well as that of discretization of the finite element method. The error estimate in the -norm is sharper than that obtained by the author [D. Koyama, Error estimates of the DtN finite element method for the exterior Helmholtz problem, J. Comput. Appl. Math. 200 (1) (2007) 21–31] for the truncated DtN boundary condition. [Copyright &y& Elsevier]

In this paper, a new space-time discretization is proposed which is based on a modified mass matrix. The mass associated with a surface layer of elements is redistributed such that the inertia at the boundary is removed. This approach is motivated by the observation that standard space-time discretization schemes applied to dynamic contact problems yield spurious oscillations. A widely used approach for the numerical simulation of these problems is based on Lagrange multipliers which represent the contact stresses. But the algebraic contact conditions in combination with the inertia volume terms often yield nonphysical results for the contact stresses, and the stability of the algorithm can be lost. Our modified matrix is calculated via nonstandard quadrature formulas that require no extra computational effort. In addition, the conservation properties of the underlying algorithm are carried over to the modified method, and the standard optimal a priori estimates are still satisfied. Numerical examples confirm the optimality of the approach and its stabilization effect applied to contact problems. [ABSTRACT FROM AUTHOR]

In this paper, we study variational discretization for the constrained optimal control problem governed by convection dominated diffusion equations, where the state equation is approximated by the edge stabilization Galerkin method. A priori error estimates are derived for the state, the adjoint state and the control. Moreover, residual type a posteriori error estimates in the the²-norm are obtained. Finally, two numerical experiments are presented to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a streamline diffusion finite element approximation scheme for the constrained optimal control problem governed by linear convection dominated diffusion equations. We prove the existence and uniqueness of the discretized scheme. Then a priori and a posteriori error estimates are derived for the state, the co-state and the control. Three numerical examples are presented to illustrate our theoretical results. AMS subject classifications: 65N30 [ABSTRACT FROM AUTHOR]

In this paper, we investigate a priori error estimates and superconvergence properties for a model optimal control problem of bilinear type, which includes some parameter estimation application. The state and co-state are discretized by piecewise linear functions and control is approximated by piecewise constant functions. We derive a priori error estimates and superconvergence analysis for both the control and the state approximations. We also give the optimal L²-norm error estimates and the almost optimal L∞-norm estimates about the state and co-state. The results can be readily used for constructing a posteriori error estimators in adaptive finite element approximation of such optimal control problems. [ABSTRACT FROM AUTHOR]

This paper is concerned with an ill-posed problem which results from the area of molecular imaging and is known as BLT problem. Using Tikhonov regularization technique, a quadratic optimization problem can be formulated. We provide an improved error estimate for the finite element approximation of the regularized optimization problem. Some numerical examples are presented to demonstrate our theoretical results. [ABSTRACT FROM AUTHOR]

The present paper is concerned with the unilateral contact model and the Coulomb friction law in linear elastostatics. We consider a mixed formulation in which the unknowns are the displacement field and the normal and tangential constraints on the contact area. The chosen finite element method involves continuous elements of degree one and continuous piecewise affine multipliers on the contact zone. A convenient discrete contact and friction condition is introduced in order to perform a convergence study. We finally obtain a first a priori error estimate under the assumptions ensuring the uniqueness of the solution to the continuous problem. [ABSTRACT FROM AUTHOR]

The explicit finite element method for transient dynamics of linear elasticity is formulated by using Galerkin method for space and the central difference method for time. An a priori error estimate is derived and the optimal rate of convergence for displacement similar to the linear elliptic problem is found. The error estimation is extended to velocity, internal (strain) energy and kinetic energy for engineering applications. The approximation error of initial data is analyzed. The error estimate is refined for a class of engineering applications with zero initial deformation and initial force. Examples of a 1-D rod axial vibration and a 2-D plate in-plane vibration are solved using linear elements as verification. [Copyright &y& Elsevier]

This paper deals with the rigorous error analysis of finite element solutions of the two- and three-dimensional nonstationary compressible Navier­Stokes equations. The streamline diffusion technique is applied to the continuity equation to obtain a stabilized finite element scheme. It is proved that the numerical formulation has a unique solution and the solution is convergent. An a priori error estimate uniform in time is obtained. [ABSTRACT FROM AUTHOR]

We introduce and analyze a new mixed discontinuous Galerkin method for approximation of an electric field. We carry out its error analysis and prove an error estimate that is optimal in the mesh size. Some numerical results are given to confirm the theoretical convergence. [ABSTRACT FROM AUTHOR]