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.