劉 雪峰

For conforming finite element approximations of the Laplacian eigenfunctions, a fully computable guaranteed error bound in the L2 norm sense is proposed. The bound is based on the a priori error estimate for the Galerkin projection of the conforming finite element method, and has an optimal speed of convergence for the eigenfunctions with the worst regularity. The resulting error estimate bounds the distance of spaces of exact and approximate eigenfunctions and, hence, is robust even in the case of multiple and tightly clustered eigenvalues. The accuracy of the proposed bound is illustrated by numerical examples.

This paper considers the finite element solution of the boundary value problem of Poisson's equation and proposes a guaranteed local error estimation based on the hypercircle method. Compared to the existing literature on qualitative error estimation, the proposed error estimation provides an explicit and sharp bound for the approximation error in the subdomain of interest, and its efficiency can be enhanced by further utilizing a non-uniform mesh. Such a result is applicable to problems without H2-regularity, since it only utilizes the first order derivative of the solution. The efficiency of the proposed method is demonstrated by numerical experiments for both convex and non-convex 2D domains with uniform or non-uniform meshes.

For compact self-adjoint operators in Hilbert spaces, two algorithms are proposed to provide fully computable a posteriori error estimate for eigenfunction approximation. Both algorithms apply well to the case of tight clusters and multiple eigenvalues, under the settings of target eigenvalue problems. Algorithm I is based on the Rayleigh quotient and the min-max principle that characterizes the eigenvalue problems. The formula for the error estimate provided by Algorithm I is easy to compute and applies to problems with limited information of Rayleigh quotients. Algorithm II, as an extension of the Davis–Kahan method, takes advantage of the dual formulation of differential operators along with the Prager–Synge technique and provides greatly improved accuracy of the estimate, especially for the finite element approximations of eigenfunctions. Numerical examples of eigenvalue problems of matrices and the Laplace operators over convex and non-convex domains illustrate the efficiency of the proposed algorithms.

In this paper, a robust and efficient algorithm is proposed to calculate the intersection points of two planar algebraic curves with guaranteed tolerance. The proposed method takes advantage of the fundamental methods in the fields of CAGD, solution verification for nonlinear equations and symbolic computation. Specifically, the subdivision method is applied to quickly exclude the regions without intersection points, and then Krawczyk's method is used to find a sharp and guaranteed bound for the intersection points. For ill-conditional cases, Sturm's theorem is applied to determine if there are any intersection points in undetermined regions. We present examples to demonstrate the robustness and efficiency of our algorithm, and comparisons with classic methods and a state-of-the-art method are also provided. The method can be easily adapted to computing the intersection points of two parametric curves.

This paper proposes a computer-assisted solution existence verification method for the stationary Navier–Stokes equation over general 3D domains. The proposed method verifies that the exact solution as the fixed point of the Newton iteration exists around the approximate solution through rigorous computation and error estimation. The explicit values of quantities required by applying the fixed-point theorem are obtained by utilizing newly developed quantitative error estimation for finite element solutions to boundary value problems and eigenvalue problems of the Stokes equation.

For the linear Lagrange interpolation over a triangular domain, we propose an efficient algorithm to rigorously evaluate the interpolation error constant under the maximum norm by using the finite-element method (FEM). In solving the optimization problem corresponding to the interpolation error constant, the maximum norm in the constraint condition is the most difficult part to process. To handle this difficulty, a novel method is proposed by combining the orthogonality of the space decomposition using the Fujino–Morley FEM space and the convex-hull property of the Bernstein representation of functions in the FEM space. Numerical results for the lower and upper bounds of the interpolation error constant for triangles of various types are presented to verify the efficiency of the proposed method.

For the Stokes equation over 2D and 3D domains, explicit a posteriori and a priori error estimation are novelly developed for the finite element solution. The difficulty in handling the divergence-free condition of the Stokes equation is solved by utilizing the extended hypercircle method along with the Scott-Vogelius finite element scheme. Since all terms in the error estimation have explicit values, by further applying the interval arithmetic and verified computing algorithms, the computed results provide rigorous estimation for the approximation error. As an application of the proposed error estimation, the eigenvalue problem of the Stokes operator is considered and rigorous bounds for the eigenvalues are obtained. The efficiency of proposed error estimation is demonstrated by solving the Stokes equation on both convex and non-convex 3D domains.

Recently, the eigenvalue problems formulated with symmetric positive definite bilinear forms have been well investigated with the aim of explicit bounds for the eigenvalues. In this paper, the existing theorems for bounding eigenvalues are further extended to deal with the case of eigenvalue problems defined by positive semi-definite bilinear forms. As an application, the eigenvalue estimation theorems are applied to the error constant estimation for polynomial projections.

The quantitative estimation for the interpolation error constants of the Fujino–Morley interpolation operator is considered. To give concrete upper bounds for the constants, which is reduced to the problem of providing lower bounds for eigenvalues of bi-harmonic operators, a new algorithm based on the finite element method along with verified computation is proposed. In addition, the quantitative analysis for the variation of eigenvalues upon the perturbation of the shape of triangles is provided. Particularly, for triangles with longest edge length less than one, the optimal estimation for the constants is provided. An online demo with source codes of the constants calculation is available at http://www.xfliu.org/onlinelab/.

To provide mathematically rigorous eigenvalue bounds for the Steklov eigenvalue problem, an enhanced version of the eigenvalue estimation algorithm developed by the third author is proposed. Compared with the existing algorithm, which deals with eigenvalue problems formulated by positive definite bilinear forms, the newly proposed method can be used to solve the problem defined with positive semidefinite bilinear forms. In practical eigenvalue estimation for the Steklov eigenvalue problem, the Crouzeix-Raviart finite element method along with quantitative error estimation is adopted. Numerical experiments for eigenvalue problems defined on a square domain and an L-shaped domain are provided to validate the precision of computed eigenvalue bounds.

The non-conforming linear (P1) triangular FEM can be viewed as a kind of the discontinuous Galerkin method, and is attractive in both the theoretical and practical purposes. Since various error constants must be quantitatively evaluated for its accurate a priori and a posteriori error estimates, we derive their theoretical upper bounds and some computational results. In particular, the Babuška-Aziz maximum angle condition is required just as in the case of the conforming P1 triangle. Some applications and numerical results are also included to see the validity and effectiveness of our analysis.

The paper develops an explicit a priori error estimate for finite element solution to nonhomogeneous Neumann problems. For this purpose, the hypercircle over finite element spaces is constructed and the explicit upper bound of the constant in the trace theorem is given. Numerical examples are shown in the final section, which implies the proposed error estimate has the convergence rate as 0.5.

An algorithm is proposed to give explicit lower bounds of the Stokes eigenvalues by utilizing two nonconforming finite element methods: Crouzeix–Raviart (CR) element and enriched Crouzeix–Raviart (ECR) element. Compared with the existing literatures which give lower eigenvalue bounds under the asymptotic condition that the mesh size is “small enough”, the proposed algorithm in this paper drops the asymptotic condition and provide explicit lower bounds even for a rough mesh. Numerical experiments are also performed to validate the theoretical results.

For the quadratic Lagrange interpolation function, an algorithm is proposed to provide explicit and verified bound for the interpolation error constant that appears in the interpolation error estimation. The upper bound for the interpolation constant is obtained by solving an eigenvalue problem along with explicit lower bound for its eigenvalues. The lower bound for interpolation constant can be easily obtained by applying the Rayleigh-Ritz method. Numerical computation is performed to demonstrate the sharpness of lower and upper bounds of the interpolation constants over triangles of different shapes. An online computing demo is available at http://www.xfliu.org/onlinelab/. (C) 2017 Elsevier Inc. All rights reserved.

For eigenvalue problems of self-adjoint differential operators, a universal framework is proposed to give explicit lower and upper bounds for the eigenvalues. In the case of the Laplacian operator, by applying Crouzeix-Raviart finite elements, an efficient algorithm is developed to bound the eigenvalues for the Laplacian defined in 1D, 2D and 3D spaces. Moreover, for nonconvex domains, for which case there may exist singularities of eigen-functions around re-entrant corners, the proposed algorithm can easily provide eigenvalue bounds. By further adopting the interval arithmetic, the explicit eigenvalue bounds from numerical computations can be mathematically correct. (C) 2015 Elsevier Inc. All rights reserved.

Traditional structural optimization methods require predefined load conditions. The resulting structure is optimal under the given condition, but can be weak under different loads. Objects can suffer from various forces in practical applications. The overall performance of objects cannot be guaranteed and thus more material than actually needed is used. In this work we propose a novel approach to enhance the global strength of 3D objects under all possible load distribution, which make the strength of the object isotropic to resist different forces. The method is based on modal analysis. We first detect the weak region of the object and then reinforce it by optimizing the eigenvalue of the stiffness matrix. Based on the concept of Rayleigh Quotient, an efficient algorithm is also presented. Experiments show that our method can effectively improve the global strength of 3D objects.

This paper proposes a verified numerical method of proving the invertibility of linear elliptic operators. This method also provides a verified norm estimation for the inverse operators. This type of estimation is important for verified computations of solutions to elliptic boundary value problems. The proposed method uses a generalized eigenvalue problem to derive the norm estimation. This method has several advantages. Namely, it can be applied to two types of boundary conditions: the Dirichlet type and the Neumann type. It also provides a way of numerically evaluating lower and upper bounds of target eigenvalues. Numerical examples are presented to show that the proposed method provides effective estimations in most cases.

For Poisson's equation over a polygonal domain of general shape, the solution of which may have a singularity around re-entrant corners, we provide an explicit a priori error estimate for the approximate solution obtained by finite element methods of high degree. The method used herein is a direct extension of the one developed in preceding paper of the second and third listed authors, which provided a new approach to deal with the singularity by using linear finite elements. In the present paper, we also give a detailed discussion of the dependency of the convergence order on solution singularities, mesh sizes and degrees of the finite element method used.

The finite element method (FEM) is applied to bound leading eigenvalues of the Laplace operator over polygonal domains. Compared with classical numerical methods, most of which can only give concrete eigenvalue bounds over special domains of symmetry, our proposed algorithm can provide concrete eigenvalue bounds for domains of arbitrary shape, even when the eigenfunction has a singularity. The problem of eigenvalue estimation is solved in two steps. First, we construct a computable a priori error estimation for the FEM solution of Poisson's problem, which holds even for nonconvex domains with reentrant corners. Second, new computable lower bounds are developed for the eigenvalues. Because the interval arithmetic is implemented throughout the computation, the desired eigenvalue bounds are expected to be mathematically correct. We illustrate several computation examples, such as the cases of an L-shaped domain and a crack domain, to demonstrate the efficiency and flexibility of the proposed method.

We consider an explicit estimation for error constants from two basic constant interpolations on triangular finite elements. The problem of estimating the interpolation constants is related to the eigenvalue problems of the Laplacian with certain boundary conditions. By adopting the Lehmann-Goerisch theorem and finite element spaces with a variable mesh size and polynomial degree, we succeed in bounding the leading eigenvalues of the Laplacian and the error constants with high precision. An online demo for the constant estimation is also available at http://www.xfliu.org/onlinelab/.

In this paper, a numerical verification method is presented for second-order semilinear elliptic boundary value problems on arbitrary polygonal domains. Based on the Newton-Kantorovich theorem, our method can prove the existence and local uniqueness of the solution in the neighborhood of its approximation. In the treatment of polygonal domains with an arbitrary shape, which gives a singularity of the solution around the re-entrant corner, the computable error estimate of a projection into the finite-dimensional function space plays an essential role. In particular, the lack of smoothness of the solution makes classical error estimates fail on nonconvex domains. By using the Hyper-circle equation, an alternative error estimate of the projection has been proposed. Additionally, a new residual evaluation method based on the mixed finite element method works well. It yields more accurate evaluation than the existing method. The efficiency of our method is shown through illustrative numerical results on several polygonal domains.

We give some fundamental results on the error constants for the piecewise constant interpolation function and the piecewise linear one over triangles. For the piecewise linear one, we mainly analyze the conforming case, but the present results also appear to be available for the non-conforming case. We obtain explicit relations for the upper bounds of the constants, and analyze dependence of such constants on the geometric parameters of triangles. In particular, we explicitly determine some special constants including the Babuska-Aziz constant, which plays an essential role in the interpolation error estimation of the linear triangular finite element. The obtained results are expected to be widely used for a priori and a posteriori error estimations in adaptive computation and numerical verification of numerical solutions based on the triangular finite elements. We also give some numerical results for the error constants and for a. posteriori estimates of some eigenvalues related to the error constants.

有限要素法の誤差評価には、様々な誤差定数が現れる。その具体的な値は決定しにくい事もあり、存在のみが論じられることが多かったが、精度保証、事後誤差評価、適応型計算など定量的な誤差評価の重要性が増してきた結果、より具体的な値が必要になっている。本研究では、適合および非適合 1 次三角形有限要素法で現れる誤差定数を解析して、有限要素解の定量的な事前誤差評価と事後誤差評価を得た。

We give some fundamental results on the error constants for the piecewise constant interpolation function and the piecewise linear one over triangles. For the piecewise linear one, we mainly analyze the conforming case, but some results are also given for the non-conforming case. We obtain explicit relations for the dependence of such error constants on the geometric parameters of triangles. In particular, we explicitly determine the Babuska-Aziz constant, which plays an essential role in the interpolation error estimation of the linear triangular finite element. The equation for determination is the transcendental equation root lambda + tan root lambda), = 0, so that the solution can be numerically obtained with desired accuracy and verification. Such highly accurate approximate values for the constant as well as estimates for other constants can be widely used for a priori and a posteriori error estimations in adaptive computation and numerical verification of finite element solutions. (c) 2007 Elsevier B.V. All rights reserved.

We explicitly determine the Babuska-Aziz constant, which plays an essential role in the interpolation error estimation of the linear triangular finite element. The equation for determination is the transcendental equation t + tan t = 0, so that the solution call be numerically obtained with desired accuracy and verification. Such highly accurate approximate values for the constant can be widely used for a priori and a posteriori error estimations in adaptive computation and/or numerical verification.

In this paper, a subdivision scheme which generalizes a surface scheme in previous papers to volume meshes is designed. The scheme exhibits significant control over shrink-age/ size of volumetric models. It also has the ability to conveniently incorporate boundaries and creases into a smooth limit shape of models. The method presented here is much simpler and easier as compared to MacCracken and Joy's. This method makes no restrictions on the local topology of meshes. Particularly, it can be applied without any change to meshes of nonmanifold topology.

We give some fundamental results on the error constants for the piecewise constant interpolation function and the piecewise linear one over triangles. We obtain explicit relotions for the dependence of such error constants on the geometric parameters of triangles. In particular, we explicitly determine the Babuska-Aziz constant, which plays an essential role in the interpolation error estimation of the linear triangular finite element The equation for determination is the transcendental equation √λ+tan √λ = 0, so that the solution can be numerically obtained with desired accuracy and verification. Such highly accurate approximate values for the constant as well as estimates for other constants can be widely used for a priori and a posteriori error estimations in adaptive computation and numerical verification of finite element solutions. .