10 citations · 19 across the 21 of their papers we have counts for
21 papers
Guaranteed Lower Eigenvalue Bounds for Spectral Galerkin Methods with Application to Schrödinger Operators
Xuefeng Liu
Spectral Galerkin methods are renowned for high-precision eigenvalue approximation, yet a rigorous lower bound obtained directly from a spectral discretisation has remained unavail…
Explicit Two-Sided Eigenvalue Bounds for Schrödinger Operators with Singular Potentials via Finite Element Method
Xuefeng Liu
We present, to the best of our knowledge, the first numerical algorithm for explicit, computable two-sided eigenvalue bounds for Schrödinger operators H = -Delta + V on R^N, N = 2,…
Sharp Dirichlet eigenvalue inequalities on triangles
Ryoki Endo, Xuefeng Liu, Phanuel Mariano
We prove sharp Dirichlet eigenvalue inequalities for planar triangles. We settle a conjecture of Laugesen and Siudeja by showing that the equilateral triangle uniquely minimizes a…
Rigorous Eigenvalue Bounds for Schrödinger Operators with Confining Potentials on
Xuefeng Liu
We propose a rigorous method for computing two-sided eigenvalue bounds of the Schrödinger operator with a confining potential on . The method combines domain…
The Second Dirichlet Eigenvalue is Simple on Every Non-equilateral Triangle, Part II: Nearly Equilateral Triangles
Ryoki Endo, Xuefeng Liu
This paper solves the open problem of the simplicity of the second Dirichlet eigenvalue for nearly equilateral triangles, offering a complete solution to Conjecture 6.47 posed by R…
Projection error-based guaranteed L2 error bounds for finite element approximations of Laplace eigenfunctions
Xuefeng Liu, Tomáš Vejchodský
For conforming finite element approximations of the Laplacian eigenfunctions, a fully computable guaranteed error bound in the norm sense is proposed. The bound is based on t…
Shape optimization for the Laplacian eigenvalue over triangles and its application to interpolation error constant estimation
Ryoki Endo, Xuefeng Liu
A computer-assisted proof is proposed for the Laplacian eigenvalue minimization problems over triangular domains under diameter constraints. The proof utilizes recently developed g…
Error constant estimation under the maximum norm for linear Lagrange interpolation
Shirley Mae Galindo, Koichiro Ike, Xuefeng Liu
For the Lagrange interpolation over a triangular domain, we propose an efficient algorithm to rigorously evaluate the interpolation error constant under the maximum norm by using t…
Guaranteed a posteriori local error estimation for finite element solutions of boundary value problems
Taiga Nakano, Xuefeng Liu
This paper considers the finite element solution of the boundary value problem of Poisson's equation and proposes a guaranteed em a posteriori local error estimation based on the h…
Computer-assisted proof for the stationary solution existence of the Navier-Stokes equation over 3D domains
Xuefeng Liu, Mitsuhiro T. Nakao, Shin'ichi Oishi
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…
Explicit a posteriori and a priori error estimation for the finite element solution of Stokes equations
Xuefeng Liu, Mitsuhiro Nakao, Chun'guang You +1
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 handlin…
Guaranteed lower eigenvalue bounds for Steklov operators using conforming finite element methods
Taiga Nakano, Qin Li, Meiling Yue +1
For the eigenvalue problem of the Steklov differential operator, by following Liu's approach, an algorithm utilizing the conforming finite element method (FEM) is proposed to provi…
Explicit a posteriori local error estimation for FEM solutions
Taiga Nakano, Xuefeng Liu
For the finite element solution of Poisson's equation, a local a posteriori error estimation based on the Hypercircle method is proposed. Even for the solution of Poisson's equatio…
Explicit eigenvalue bounds of differential operators defined by symmetric positive semi-definite bilinear forms
Xuefeng Liu
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 thi…
Fully computable a posteriori error bounds for eigenfunctions
Xuefeng Liu, Tomáš Vejchodský
For compact self-adjoint operators in Hilbert spaces, two algorithms are proposed to provide fully computable a posteriori error estimate for eigenfunction approximation. Both algo…
Optimal estimation for the Fujino-Morley interpolation error constants
Shih-Kang Liao, Yu-Chen Shu, Xuefeng Liu
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…
Guaranteed eigenvalue bounds for the Steklov eigenvalue problem
Chun'guang You, Hehu Xie, Xuefeng Liu
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…
Explicit Estimation of Error Constants Appearing in Non-conforming Linear Triangular Finite Element
Xuefeng Liu, Fumio Kikuchi
The non-conforming linear () triangular FEM can be viewed as a kind of the discontinuous Galerkin method, and is attractive in both theoretical and practical senses. Since var…
Explicit Finite Element Error Estimates for Nonhomogeneous Neumann problems
Qin Li, Xuefeng Liu
The paper develops an explicit a priori error estimate for finite element solution to nonhomogeneous Neumann problems. For this purpose, the hypercircle equation over finite elemen…
Explicit Bound for Quadratic Lagrange Interpolation Constant on Triangular Finite Elements
Xuefeng Liu, Chun'guang You
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 interpol…
Verified eigenvalue evaluation for Laplacian over polygonal domain of arbitrary shape
Xuefeng Liu, Shin'ichi Oishi
The finite element method(FEM) is applied to bound leading eigenvalues of Laplace operator over polygonal domain. Compared with classical numerical methods, most of which can only…