Error Estimates For Approximation Research Articles

A stable weak Galerkin finite element method for Stokes problem

We study the weak Galerkin finite element method for Stokes problem. A new weak Galerkin finite element velocity–pressure space pair is presented which satisfies the discrete inf–sup condition. Based on this space pair, we establish a stable weak Galerkin approximation scheme without adding any stability term or penalty term. Then, we further derive the optimal error estimates for velocity and pressure approximations, respectively. Numerical experiments are provided to illustrate the theoretical analysis.

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.

Arnoldi-based model order reduction for linear systems with inhomogeneous initial conditions

In this paper, we discuss the Arnoldi-based model order reduction (MOR) methods for linear systems with inhomogeneous initial conditions, and present a time domain MOR method by general orthogonal polynomials. The basic procedure is to use the expansion coefficient matrix in the orthogonal polynomial space satisfying a simple recurrence formula to generate a projection matrix, which is produced by a modified Arnoldi algorithm. Then, the resulting reduced model matches a desired number of the expansion coefficients of the original system. The approximate error estimate of the reduced model is given. Since the initial conditions are well represented by the subspaces constructed by our algorithm, it can well deal with those systems with inhomogeneous initial conditions. Two benchmark examples in real applications are simulated to illustrate the effectiveness of the proposed method.

English

Compared to conforming P1 finite elements, nonconforming P1 finite element discretizations are thought to be less sensitive to the appearance of distorted triangulations. E.g., optimal-order discrete H1 norm best approximation error estimates for H2 functions hold for arbitrary triangulations. However, the constants in similar estimates for the error of the Galerkin projection for second-order elliptic problems show a dependence on the maximum angle of all triangles in the triangulation. We demonstrate on an example of a special family of distorted triangulations that this dependence is essential, and due to the deterioration of the consistency error. We also provide examples of sequences of triangulations such that the nonconforming P1 Galerkin projections for a Poisson problem with polynomial solution do not converge or converge at arbitrarily low speed. The results complement analogous findings for conforming P1 finite elements.

Hp-Adaptive Galerkin Time Stepping Methods for Nonlinear Initial Value Problems

This work is concerned with the derivation of an a posteriori error estimator for Galerkin approximations to nonlinear initial value problems with an emphasis on finite-time existence in the context of blow-up. The structure of the derived estimator leads naturally to the development of both h and hp versions of an adaptive algorithm designed to approximate the blow-up time. The adaptive algorithms are then applied in a series of numerical experiments, and the rate of convergence to the blow-up time is investigated.

The fictitious domain method with H1-penalty for the Stokes problem with Dirichlet boundary condition

We consider the fictitious domain method with H1-penalty for the Stokes problem with Dirichlet boundary condition. First, for the continuous penalty problem, we obtain the optimal error estimate O(ϵ) for both the velocity and pressure, where ϵ is the penalty parameter. Moreover, we investigate the Hm-regularity for the solution of the penalty problem. Then, we apply the finite element method with the P1/P1 element to the penalty problem. Since the solution to the penalty problem has a jump in the traction vector, we introduce some interpolation/projection operators, as well as an inf-sup condition with the norm depending on ϵ. With the help of these preliminaries, we derive the error estimates for the finite element approximation. The theoretical results are verified by the numerical experiments.

Interior Penalty Discontinuous Galerkin Methods for Second Order Linear Non-divergence Form Elliptic PDEs

This paper develops interior penalty discontinuous Galerkin (IP-DG) methods to approximate $$W^{2,p}$$ strong solutions of second order linear elliptic partial differential equations (PDEs) in non-divergence form with continuous coefficients. The proposed IP-DG methods are closely related to the IP-DG methods for advection-diffusion equations, and they are easy to implement on existing standard IP-DG software platforms. It is proved that the proposed IP-DG methods have unique solutions and converge with optimal rate to the $$W^{2,p}$$ strong solution in a discrete $$W^{2,p}$$ -norm. The crux of the analysis is to establish a DG discrete counterpart of the Calderon–Zygmund estimate and to adapt a freezing coefficient technique used for the PDE analysis at the discrete level. To obtain such a crucial estimate, we need to establish broken $$W^{1,p}$$ -norm error estimates for IP-DG approximations of constant coefficient elliptic PDEs, which is also of independent interest. Numerical experiments are provided to gauge the performance of the proposed IP-DG methods and to validate the theoretical convergence results.

Computable Error Estimates for a Nonsymmetric Eigenvalue Problem

AbstractWe provide some computable error estimates in solving a nonsymmetric eigenvalue problem by general conforming finite element methods on general meshes. Based on the complementary method, we first give computable error estimates for both the original eigenfunctions and the corresponding adjoint eigenfunctions, and then we introduce a generalised Rayleigh quotient to deduce a computable error estimate for the eigenvalue approximations. Some numerical examples are presented to illustrate our theoretical results.

Limitations of shallow nets approximation

In this paper, we aim at analyzing the approximation abilities of shallow networks in reproducing kernel Hilbert spaces (RKHSs). We prove that there is a probability measure such that the achievable lower bound for approximating by shallow nets can be realized for all functions in balls of reproducing kernel Hilbert space with high probability, which is different with the classical minimax approximation error estimates. This result together with the existing approximation results for deep nets shows the limitations for shallow nets and provides a theoretical explanation on why deep nets perform better than shallow nets.

Error estimation and adaptive moment hierarchies for goal-oriented approximations of the Boltzmann equation

This paper presents an a-posteriori goal-oriented error analysis for a numerical approximation of the steady Boltzmann equation based on a moment-system approximation in velocity dependence and a discontinuous Galerkin finite-element (DGFE) approximation in position dependence. We derive computable error estimates and bounds for general target functionals of solutions of the steady Boltzmann equation based on the DGFE moment approximation. The a-posteriori error estimates and bounds are used to guide a model adaptive algorithm for optimal approximations of the goal functional in question. We present results for one-dimensional heat transfer and shock structure problems where the moment model order is refined locally in space for optimal approximation of the heat flux.

Finite element quasi-interpolation and best approximation

This paper introduces a quasi-interpolation operator for scalar- and vector-valued finite element spaces constructed on affine, shape-regular meshes with some continuity across mesh interfaces. This operator gives optimal estimates of the best approximation error in any Lp-norm assuming regularity in the fractional Sobolev spaces Wr,p, where p ∈ [ 1,∞ ] and the smoothness index r can be arbitrarily close to zero. The operator is stable in L1, leaves the corresponding finite element space point-wise invariant, and can be modified to handle homogeneous boundary conditions. The theory is illustrated on H1-, H(curl)- and H(div)-conforming spaces.

A priori error estimates of finite volume method for nonlinear optimal control problem

In this paper, we study a priori error estimates for the finite volume element approximation of nonlinear optimal control problem. The schemes use discretizations based on a finite volume method. For the variational inequality, we use the method of the variational discretization concept to obtain the control. Under some reasonable assumptions, we obtain some optimal order error estimates. The approximate order for the state, costate and control variables is O(h 2) or $$O\left( {{h^2}\sqrt {\left| {\ln h} \right|} } \right)$$ in the sense of L 2-norm or L ∞-norm. A numerical experiment is presented to test the theoretical results. Finally, we give some conclusions and future works.

$$\mathbf {L^\infty }$$ L ∞ -error estimates for the obstacle problem revisited

In this paper, we present an alternative approach to a priori \(L^\infty \)-error estimates for the piecewise linear finite element approximation of the classical obstacle problem. Our approach is based on stability results for discretized obstacle problems and on error estimates for the finite element approximation of functions under pointwise inequality constraints. As an outcome, we obtain the same order of convergence proven in several works before. In contrast to prior results, our estimates can, for example, also be used to study the situation where the function space is discretized but the obstacle is not modified at all.

Finite Element Exterior Calculus for Evolution Problems

Arnold, Falk, and Winther [Bull. Amer. Math. Soc. 47 (2010), 281--354] showed that mixed variational problems, and their numerical approximation by mixed methods, could be most completely understood using the ideas and tools of Hilbert complexes. This led to the development of the Finite Element Exterior Calculus (FEEC) for a large class of linear elliptic problems. More recently, Holst and Stern [Found. Comp. Math. 12:3 (2012), 263--293 and 363--387] extended the FEEC framework to semi-linear problems, and to problems containing variational crimes, allowing for the analysis and numerical approximation of linear and nonlinear geometric elliptic partial differential equations on Riemannian manifolds of arbitrary spatial dimension, generalizing surface finite element approximation theory. In this article, we develop another distinct extension to the FEEC, namely to parabolic and hyperbolic evolution systems, allowing for the treatment of geometric and other evolution problems. Our approach is to combine the recent work on the FEEC for elliptic problems with a classical approach to solving evolution problems via semi-discrete finite element methods, by viewing solutions to the evolution problem as lying in time-parameterized Hilbert spaces (or Bochner spaces). Building on classical approaches by Thomee for parabolic problems and Geveci for hyperbolic problems, we establish a priori error estimates for Galerkin FEM approximation in the natural parametrized Hilbert space norms. In particular, we recover the results of Thomee and Geveci for two-dimensional domains and lowest-order mixed methods as special cases, effectively extending their results to arbitrary spatial dimension and to an entire family of mixed methods. We also show how the Holst and Stern framework allows for extensions of these results to certain semi-linear evolution problems.

Rational Operators Based on q-Integers

Shepard-type rational operators based on q-integers are studied and convergence results and pointwise approximation error estimates improving previous statements are obtained. Influence of choice of q on the error estimates is also discussed. Techniques in CAGD for shape modeling of rational curves based on above operators are also presented and numerical examples are given.

Finite element approximation of Dirichlet control using boundary penalty method for unsteady Navier–Stokes equations

This paper is concerned with the analysis of the finite element approximations of Dirichlet control problem using boundary penalty method for unsteady Navier–Stokes equations. Boundary penalty method has been used as a computationally convenient approach alternative to Dirichlet boundary control problems governed by Navier−Stokes equations due to its variational properties. Analysis of the mixed Galerkin finite element method applied to the spatial semi-discretization of the optimality system, from which optimal control can be computed, is presented. An optimal L ∞ (L 2 ) error estimate of the numerical approximations of the optimality system is derived. Feasibility and applicability of the approach are illustrated by numerically solving a canonical flow control problem.

Steklov approximations of harmonic boundary value problems on planar regions

Error estimates for approximations of solutions of Laplace’s equation with Dirichlet, Robin or Neumann boundary value conditions are described. The solutions are represented by orthogonal series using the harmonic Steklov eigenfunctions. Error bounds for partial sums involving the lowest eigenfunctions are found. When the region is a rectangle, explicit formulae for the Steklov eigenfunctions and eigenvalues are known. These were used to find approximations for problems with known explicit solutions. Results about the accuracy of these solutions, as a function of the number of eigenfunctions used, are given.

A priori L∞(L2) error estimates for finite element approximations to the wave equation with interface

In this article a fitted finite element method is proposed and analyzed for wave equation with discontinuous coefficients. Typical semidiscrete and an implicit fully discrete schemes are presented and analyzed. Optimal a priori error estimates for both semidiscrete and fully discrete schemes are proved in L∞(L2) norm. The convergence analysis relies heavily on time reconstructions of continuous and discrete solutions, in conjunction with some known results on elliptic interface problems. Finally, a numerical experiment is presented to verify our theoretical result.

Anisotropic [formula omitted]-mesh optimization technique based on the continuous mesh and error models

We develop a new mesh adaptive technique for the numerical solution of partial differential equations (PDEs) using the hp-version of the finite element method (hp-FEM). The technique uses a combination of approximation and interpolation error estimates to generate anisotropic triangular elements as well as appropriate polynomial approximation degrees. We present a hp-version of the continuous mesh model as well as the continuous error model which are used for the formulation of a mesh optimization problem. Solving the optimization problem leads to hp-mesh with the smallest number of degrees of freedom, under the constraint that the approximate solution has an error estimate below a given tolerance. Further, we propose an iterative algorithm to find a suitable anisotropic hp-mesh in the sense of the mesh optimization problem. Several numerical examples demonstrating the efficiency and applicability of the new method are presented.

A Goal-Oriented Error Estimator for a Class of Homogenization Problems

We present a goal-oriented a posteriori error estimator for finite element approximations of a class of homogenization problems. As a rule, homogenization problems are defined through the coupling of a macroscopic solution and the solution of auxiliary problems. In this work we assume that the homogenized problem is known and that it depends on a finite number of auxiliary problems. The accuracy in the goal functional depends therefore on the discretization error of the macroscopic and the auxiliary solutions. We show that it is possible to compute the error contributions of all solution components separately and use this information to balance the different discretization errors. Additionally, we steer a local mesh refinement for both the macroscopic problem and the auxiliary problems. The high efficiency of this approach is shown by numerical examples. These include the upscaling of a periodic diffusion tensor, the case of a Stokes flow over a porous bed, and the homogenization of a fuel cell model which includes the flow in a gas channel over a porous substrate coupled with a multispecies nonlinear transport equation.