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In the context of unfitted finite element discretizations the realization of high order methods is challenging due to the fact that the geometry approximation has to be sufficiently accurate. Recently a new unfitted finite element method was introduced which achieves a high order approximation of the geometry for domains which are implicitly described by smooth level set functions. This method is based on a parametric mapping which transforms a piecewise planar interface (or surface) reconstruction to a high order approximation. In the paper [C. Lehrenfeld and A. Reusken, IMA J. Numer. Anal. 38 (2018), No. 3, 1351–1387] an a priori error analysis of the method applied to an interface problem is presented. The analysis reveals optimal order discretization error bounds in the H1-norm. In this paper we extend this analysis and derive optimal L2-error bounds. [ABSTRACT FROM AUTHOR]

A decoupled finite element method with different time steps for the nonstationary Darcy--Brinkman problem is considered in this paper. Moreover, for the presented method, the stability analysis and error estimates are deduced. Finally, numerical tests are provided that demonstrate the efficiency of the method. It is found the presented method can save lots of computational time compared with standard coupled method. [ABSTRACT FROM AUTHOR]

In this paper an application of the homotopy analysis method for some type of nonlinear and linear integral equations of the second kind is presented. A special case of considered equation is the Volterra- Fredholm integral equation. In homotopy analysis method a series is created. It has shown that if the series is convergent, its sum is the solution of the considered equation. It has been also shown that under proper assumptions the considered equation possesses a unique solution and the series obtained in homotopy analysis method is convergent. The error of the approximate solution was estimated. This approximate solution is obtained when we limit to the partial sum of the series.Application of the method is illustrated with examples. [ABSTRACT FROM AUTHOR]

It is still an open problem to prove a priori error estimates for finite volume schemes of higher order MUSCL type, including limiters, on unstructured meshes, which show some improvement compared to first order schemes. In this paper we use these higher order schemes for the discretization of convection dominated elliptic problems in a convex bounded domain Ω in ℝ2 and we can prove such kind of an a priori error estimate. In the part of the estimate, which refers to the discretization of the convective term, we gain h1/2. Although the original problem is linear, the numerical problem becomes nonlinear, due to MUSCL type reconstruction/limiter technique. [ABSTRACT FROM AUTHOR]

This paper derives some interior superconvergence estimates for finite element solutions of the Stokes problem by a local L2 projection method. The results depend only on local properties of the Stokes problem and the finite element approximations. [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]

In this paper we study the convergence of Halley's method as a method for finding all zeros of a polynomial simultaneously. We present two types of local convergence theorems as well as a semilocal convergence theorem for Halley's method for simultaneous computation of polynomial zeros. [ABSTRACT FROM AUTHOR]

It is well-known that the remaining term of a classical n-point Newton-Cotes quadrature depends on at least an n-order derivative of the integrand function. Discounting the fact that computing an n-order derivative requires a lot of differentiation for large n, the main problem is that an error bound for an n-point Newton-Cotes quadrature is only relevant for a function that is n times differentiable, a rather stringent condition. In this paper, by defining two specific linear kernels, we resolve this problem and obtain new error bounds for all closed and open types of Newton-Cotes quadrature rules. The advantage of the obtained bounds is that they do not depend on the norms of the integrand function and are very general such that they cover almost all existing results in the literature. Some illustrative examples are given in this direction. [ABSTRACT FROM AUTHOR]

-The main purpose of this paper is to propose the Chebyshev spectral-collocation method for a class of the weakly singular Volterra integral equations (VIEs) with proportional delay. The proposed method also are applicable to a class of the weakly singular VIEs with proportional delay possessing unsmooth solution. To provide a rigorous error analysis for the proposed method, we prove the the uniqueness and smoothness of the solution. The error analysis shows that the numerical errors decay exponentially in the infinity norm and the Chebyshev weighted Hilbert space norms. Numerical results are presented to confirm the theoretical prediction of the exponential rate of convergence. [ABSTRACT FROM AUTHOR]

- This paper presents the multiharmonic analysis of a distributed parabolic optimal control problem in a time-periodic setting. We prove the existence and uniqueness of the solution of some weak space-time variational formulation for the parabolic time-periodic boundary value problem appearing in the constraints for the optimal control problem. Since the cost functional is quadratic, the optimal control problem is uniquely solvable as well. In order to solve the optimal control problem numerically, we state its optimality system and discretize it by the multiharmonic finite element method leading to a system of linear algebraic equations which decouples into smaller systems. We construct preconditioners for these systems which yield robust convergence rates and optimal complexity for the preconditioned minimal residual method. All systems can be solved totally in parallel. Furthermore, we present a complete analysis for the error introduced by the multiharmonic finite element discretization as well as some numerical results confirming our theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper a general parabolic problem is considered and discretized by the discontinuous Galerkin (DG) method in time and, in general, in space. Optimal a priori error estimates in space as well as in time are derived and applied to the heat equation and to a nonlinear convection-diffusion equation. [ABSTRACT FROM AUTHOR]

In this paper we use the discontinuous Galerkin finite element method for the space-semidiscretization of a nonlinear nonstationary convection--diffusion problem defined on a nonpolygonal two-dimensional domain. Using Zláámal's concept of the ideal curved elements, we define a finite element space . We prove the ''ideal'' versions of the inverse and the multiplicative trace inequalities known for standard straight triangulations. Further, we define a projection on the finite element space and study its approximation properties. The obtained results allow us to derive an H1-optimal error estimate for the discontinuous Galerkin method employing the ideal curved elements. [ABSTRACT FROM AUTHOR]

This paper develops a combined a posteriori analysis for the discretization and iteration errors in the solution of elliptic eigenvalue problems by the finite element method. The emphasis is on the iterative solution of the discretized eigenvalue problem by a Krylov-space method. The underlying theoretical framework is that of the Dual Weighted Residual (DWR) method for goal-oriented error estimation. On the basis of computable a posteriori error estimates the algebraic iteration can be adjusted to the discretization within a successive mesh adaptation process. The functionality of the proposed method is demonstrated by numerical examples. [ABSTRACT FROM AUTHOR]

A unified framework for a residual-based a posteriori error analysis of standard conforming finite element methods as well as non-standard techniques such as nonconforming and mixed methods has been developed in [Carstensen, Numer. Math. 100: 617 – 637, 2005, Carstensen, Gudi, and Jensen, A unifying theory of a posteriori error control for discontinuous Galerkin FEM, Department of Mathematics, Humboldt University of Berlin, 2008, Carstensen and Hoppe, J. Numer. Math. 13: 19 – 32, 2005, Carstensen and Hu, Numer. Math. 107: 473 – 502, 2007, Carstensen, Hu, and Orlando, SIAM J. Numer. Anal. 45: 68 – 82, 2007]. This paper provides such a framework for an a posteriori error control of nonconforming finite element discretizations of H(curl)-elliptic problems as they arise from low-frequency electromagnetics. These nonconforming approximations include the interior penalty discontinuous Galerkin (IPDG) approach considered in [Houston, Perugia, and Schötzau, SIAM J. Numer. Anal. 42: 434 – 459, 2004, Houston, Perugia, and Schötzau, IMA J. Numer. Anal. 27: 122 – 150, 2007], and mortar edge element approximations studied in [Belgacem, Buffa, and Maday, SIAM J. Numer. Anal. 39: 880 – 901, 2001, Hoppe, East-West J. Numer. Math. 7: 159 – 173, 1999, Hoppe, Adaptive domain decomposition techniques in electromagnetic field computation and electrothermomechanical coupling problems: Springer, 2002, Hoppe, J. Comp. Appl. Math. 168: 245 – 254, 2004, Hoppe, Contemporary Math. 383, 63 – 111, 2005, Rapetti, Buffa, Maday, and Bouillault, COMPEL 19: 332 – 340, 2000, Xu and Hoppe, SIAM J. Numer. Anal. 43: 1276 – 1294, 2005]. [ABSTRACT FROM AUTHOR]

In [2] we introduced a new type of mixed finite element approximations for two- and three-dimensional problems on distorted polygonal and polyhedral meshes that consist of cells having different forms. Additional degrees of freedom that arise in the process are excluded by a special condition that is natural for the mixed finite element approximations considered. This paper is devoted to the error analysis of the respective finite element solutions. We show that under certain assumptions on the regularity of the exact solution the convergence rate for the new approximations is the same as for the Raviart-Thomas finite element approximations of the lowest order. [ABSTRACT FROM AUTHOR]

In this paper a duality-based a posteriori error analysis is developed for the conforming hp Galerkin finite element approximation of second-order elliptic problems Duality arguments combined with Galerkin orthogonalty yield representations of the error in arbitrary quantities of interest From these error estimates, criteria are derived for the simultaneous adaptation of the mesh size h and the polynomial degree p. The effectivity of this procedure is confirmed by numerical tests. [ABSTRACT FROM AUTHOR]

Presents a second-order scheme for singularly perturbed reaction-diffusion equations with discontinuous source term. Overview of the scheme and a general framework for estimating pointwise errors; Error analysis of problem for Shishkin and Bakhvalov-Shishkin Meshes; Generalization of the problem.

In this article, we discuss error estimates for nonlinear parabolic problems using discontinuous Galerkin methods which include HDG method in the spatial direction while keeping time variable continuous. When piecewise polynomials of degree k ⩾ 1 are used to approximate both the potential as well as the flux, it is shown that the error estimate for the semi-discrete flux in L∞(0, T; L2)-norm is of order k + 1. With the help of a suitable post-processing of the semi-discrete potential, it is proved that the resulting post-processed potential converges with order of convergence O ( log ⁡ (T / h 2) h k + 2 ) $\begin{array}{} \displaystyle O\big(\!\sqrt{{}\log(T/h^2)}\,h^{k+2}\big) \end{array}$ in L∞(0, T; L2)-norm. These results extend the HDG analysis of Chabaud and Cockburn [Math. Comp. 81 (2012), 107–129] for the heat equation to non-linear parabolic problems. [ABSTRACT FROM AUTHOR]

A Neumann boundary control problem for a second order elliptic state equation is considered. The problem is regularized by an energy term which is equivalent to the H- 1/2( Γ) -norm of the control. Both the unconstrained and the control constrained cases are investigated. The regularity of the state, control, and co-state variables is studied with particular focus on the singularities due to the corners of the two-dimensional domain. The state and co-state are approximated by piecewise linear finite elements. For the approximation of the control variable we use carefully designed spaces of piecewise linear or piecewise constant functions, such that an inf-sup condition is satisfied. Error estimates for the approximate solution are proved for all three variables and we show a relation between convergence rate and the opening angles at corners of the domain. As the control grows in general unboundedly near the concave corners for unconstrained problems, it becomes active and hence regular when control constraints are present. We show that in this case the convergence rates are higher than in the unbounded case. Numerical tests suggest that the estimates derived are optimal in the unconstrained case but too pessimistic in the control constrained case. [ABSTRACT FROM AUTHOR]

We present the error analysis of a particular Oldroyd-B type model with the limiting Weissenberg number going to infinity. Assuming a suitable regularity of the exact solution we study the error estimates of a standard finite element method and of a combined finite element/finite volume method. Our theoretical result shows first order convergence of the finite element method and the error of the order 퓞( h3/4) for the finite element/finite volume method. These error estimates are compared and confirmed by the numerical experiments. [ABSTRACT FROM AUTHOR]

The coupled Stokes and Darcy equations are approximated by a strongly conservative finite element method. The discrete spaces are the divergence-conforming velocity space with matching pressure space such as the Raviart-Thomas spaces. This work proves optimal error estimate of the velocity in the L2 norm in the domain and on the interface. Lipschitz regularity of the interface is sufficient to obtain the results. [ABSTRACT FROM AUTHOR]

We consider a conforming finite element approximation of the Reissner-Mindlin eigenvalue system, for which a robust a posteriori error estimator for the eigenvector and the eigenvalue errors is proposed. For that purpose, we first perform a robust a priori error analysis without strong regularity assumption. Upper and lower bounds are then obtained up to higher order terms that are superconvergent, provided that the eigenvalue is simple. The convergence rate of the proposed estimator is confirmed by a numerical test. [ABSTRACT FROM AUTHOR]

This article continues prior work of the authors on the combined a posteriori analysis for the discretization and iteration errors in the finite element approximation of linear elliptic problems to the nonlinear case. The underlying theoretical framework is again that of the Dual Weighted Residual (DWR) method for goal-oriented error control. The accuracy in the algebraic solution process can be balanced with that due to discretization using computable a posteriori error estimates in which the outer nonlinear and inner linear iteration errors are separated from the discretization error. This results in effective stopping criteria for the algebraic iteration, which are elaborated particularly for Newton-type methods. The performance of the proposed strategies is demonstrated for several nonlinear test problems. [ABSTRACT FROM AUTHOR]

We consider an adaptive finite element method (AFEM) for the Laplace eigenvalue problem in bounded polygonal or polyhedral domains. We provide an a posteriori error analysis based on a residual type estimator which consists of element and face residuals. The a posteriori error analysis further involves an oscillation term. We prove a reduction in the energy norm of the discretization error and the oscillation term. Numerical results are given illustrating the performance of the AFEM. [ABSTRACT FROM AUTHOR]

We provide an a posteriori error analysis of finite element approximations of pointwise state constrained distributed optimal control problems for second order elliptic boundary value problems. In particular, we derive a residual-type a posteriori error estimator and prove its efficiency and reliability up to oscillations in the data of the problem and a consistency error term. In contrast to the case of pointwise control constraints, the analysis is more complicated, since the multipliers associated with the state constraints live in measure spaces. The analysis essentially makes use of appropriate regularizations of the multipliers both in the continuous and in the discrete regime. Numerical examples are given to illustrate the performance of the error estimator. [ABSTRACT FROM AUTHOR]

We devise an indirect multiple shooting (MS) method for PDE constrained optimization with a posteriori error estimates and local mesh refinement. MS is applied to the KKT system as is exemplified for a linear quadratic optimal control problem. The setup of MS in function spaces and their discrete analogs is discussed. Error representation formulas based on Galerkin orthogonality are derived. They involve sensitivity analysis by an adjoint problem and employ standard error representation on subintervals combined with additional projection errors at the shooting nodes. A posteriori error estimates and mesh refinement indicators are derived from this error representation. We discuss several mesh structures originating from different restrictions to local refinement. Finally, we present numerical results for the heat equation and problems exhibiting simple dynamics. In a second article, we will address the embedding of the scheme into a nonlinear shooting method. [ABSTRACT FROM AUTHOR]

We develop an adaptive finite element concept for elliptic optimal control problems with constraints on the state. In order to generate goal-oriented meshes, we extend the Dual Weighted Residual (DWR) concept proposed by Becker and Rannacher for PDE-constrained optimization to the state constrained case. Using the optimality system of the underlying PDE-constrained optimal control problem we obtain a representation for the error in the objectives. Based on this representation we define local error indicators, whose performance we investigate by means of a numerical example. [ABSTRACT FROM AUTHOR]

We analyze existing discontinuous Galerkin methods on quasi-uniform meshes for singularly perturbed problems. We prove weighted L2 error estimates. We use the weighted estimates to prove L2 error estimates in regions where the solution is smooth. We also prove pointwise estimates in these regions. [ABSTRACT FROM AUTHOR]

The subject-matter is the analysis of the discontinuous Galerkin finite element method of lines applied to a linear nonstationary convection-diffusion-reaction problem. In the contrary to the standard FEM the requirement of the conforming properties is omitted. The discretization is carried out with respect to space variables, whereas time remains continuous. In the discontinuous Galerkin discretization, the nonsymmetric stabilization of diffusion terms combined with interior and boundary penalty is applied. In the evaluation of fluxes the idea of upwinding is used. This allows to obtain an optimal error estimate, also verified by numerical experiments. [ABSTRACT FROM AUTHOR]

Hp-adaptive finite element methods require error estimates of the solution at the current order and one order higher. Hierarchical-based estimation strategies have proved effective in computing errors at the current order for nonlinear parabolic equations. Recently a new approach, interpolation error-based (IEB) error estimation, for constructing a posteriori error estimates at both orders has been developed for linear reaction-diffusion equations. The main results are: (i) IEB error estimation can be applied to nonlinear reaction-diffusion equations in one space dimension; (ii) the hierarchical estimator is an implicit IEB method and thus, works for reaction-diffusion problems; (iii) a hierarchical extension for computing higher-order error estimates is asymptotically exact. Computational results illustrating the theory and comparing the implicit (hierarchical) strategy with the earlier explicit IEB methods are presented. [ABSTRACT FROM AUTHOR]

We study the discretization accuracy for first-order system least squares (FOSLS) applied to Poisson's equation as a model problem. The FOSLS formulation is based on an H¹ elliptic bilinear form F. Since the order of convergence of the discretization in the L² and H¹ norms depends on the regularity of F, we examine this property in detail. We then use these results together with an Aubin-Nitsche bound to develop improved discretization error estimates. [ABSTRACT FROM AUTHOR]