The scaled boundary finite element method (SBFEM) is a relatively recent boundary element method that allows the approximation of solutions to PDEs without the need of a fundamental solution. A theoretical framework for the convergence analysis of SBFEM is proposed here. This is achieved by defining a space of semi-discrete functions and constructing an interpolation operator onto this space. We prove error estimates for this interpolation operator and show that optimal convergence to the solution can be obtained in SBFEM. These theoretical results are backed by a numerical example.
Displacement based finite element formulations over polygons: a comparison between Laplace interpolants, strain smoothing and scaled boundary polygon formulation
School of Civil & Environmental Engineering The University of New South Wales（新南威尔士大学土木与环境工程学院）
In this paper, we prove the exponential convergence of the multipole and local expansions, shifting and translation operators used in fast multipole methods (FMMs) for 3-dimensional Laplace equations in layered media. These theoretical results ensure the exponential convergence of the FMM which has been shown by the numerical results recently reported in [9]. As the free space components are calculated by the classic FMM, this paper will focus on the analysis for the reaction components of the Green's function for the Laplace equation in layered media. We first prove that the density functions…
We present an error analysis for the discontinuous Galerkin method applied to the discrete-ordinate discretization of the steady-state radiative transfer equation. Under some mild assumptions, we show that the DG method converges uniformly with respect to a scaling parameter $\varepsilon$ which characterizes the strength of scattering in the system. However, the rate is not optimal and can be polluted by the presence of boundary layers. In one-dimensional slab geometries, we demonstrate optimal convergence when boundary layers are not present and analyze a simple strategy for balance interior …
One of the main tools for solving linear systems arising from the discretization of the Helmholtz equation is the shifted Laplace preconditioner, which results from the discretization of a perturbed Helmholtz problem $-Δu - (k^2 + i \varepsilon )u = f$ where $0 \neq \varepsilon \in \mathbb{R}$ is an absorption parameter. In this work we revisit the idea of combining the shifted Laplace preconditioner with two-level deflation and apply it to Helmholtz problems discretized with linear finite elements. We use the convergence theory of GMRES based on the field of values to prove that GMRES applied…
The Bayesian approach to inverse problems provides a rigorous framework for the incorporation and quantification of uncertainties in measurements, parameters and models. We are interested in designing numerical methods which are robust w.r.t. the size of the observational noise, i.e., methods which behave well in case of concentrated posterior measures. The concentration of the posterior is a highly desirable situation in practice, since it relates to informative or large data. However, it can pose a computational challenge for numerical methods based on the prior or reference measure. We prop…
High-Order AFEM for the Laplace-Beltrami Operator: Convergence Rates
Texas A&M University（德克萨斯农工大学）；Universidad de Salamanca（萨拉曼卡大学）；Chulalongkorn University（朱拉隆功大学）；CONICET（国家科学研究委员会）；Universidad Nacional del Litoral（国立里奥斯特拉大学）；University of Maryland（马里兰大学）
In this paper, applied strictly monotonic increasing scaled maps, a kind of well-conditioned linear barycentric rational interpolations are proposed to approximate functions of singularities at the origin, such as $x^α$ for $α\in (0,1)$ and $\log(x)$. It just takes $O(N)$ flops and can achieve fast convergence rates with the choice the scaled parameter, where $N$ is the maximum degree of the denominator and numerator. The construction of the rational interpolant couples rational polynomials in the barycentric form of second kind with the transformed Jacobi-Gauss-Lobatto points. Numerical exper…
This work is devoted to the development and analysis of a linearization algorithm for microscopic elliptic equations, with scaled degenerate production, posed in a perforated medium and constrained by the homogeneous Neumann-Dirichlet boundary conditions. This technique plays two roles: to guarantee the unique weak solvability of the microscopic problem and to provide a fine approximation in the macroscopic setting. The scheme systematically relies on the choice of a stabilization parameter in such a way as to guarantee the strong convergence in $H^1$ norm for both the microscopic and macrosco…
For $h>0$ and positive integers $m$, $d$, such that $m>d/2$, we study non-stationary interpolation at the points of the scaled grid $h\mathbb{Z}^d$ via the Matérn kernel $Φ_{m,d}$---the fundamental solution of $(1-Δ)^m$ in $\mathbb{R}^d$. We prove that the Lebesgue constants of the corresponding interpolation operators are uniformly bounded as $h\to0$ and deduce the convergence rate $O(h^{2m})$ for the scaled interpolation scheme. We also provide convergence results for approximation with Matérn and related compactly supported polyharmonic kernels.
An isogeometric approach for solving the Laplace-Beltrami equation on a two-dimensional manifold embedded in three-dimensional space using a Galerkin method based on Catmull-Clark subdivision surfaces is presented and assessed. The scalar-valued Laplace-Beltrami equation requires only C0 continuity and is adopted to elucidate key features and properties of the isogeometric method using Catmull-Clark subdivision surfaces. Catmull-Clark subdivision bases are used to discretise both the geometry and the physical field. A fitting method generates control meshes to approximate any given geometry wi…
We consider numerical approximations of spectral fractional Laplace-Beltrami problems on closed surfaces. The proposed numerical algorithms rely on their Balakrishnan integral representation and consist of a sinc quadrature coupled with standard finite element methods for parametric surfaces. Possibly up to a log term, optimal rates of convergence are observed and derived analytically when the discrepancies between the exact solution and its numerical approximations are measured in $L^2$ and $H^1$. The performances of the algorithms are illustrated in different settings including the approxima…
In this paper, a unified family, for any $n\geqslant 2$ and $1\leqslant k\leqslant n-1$, of nonconforming finite element schemes are presented for the primal weak formulation of the $n$-dimensional Hodge-Laplace equation on $HΛ^k\cap H^*_0Λ^k$ and on the simplicial subdivisions of the domain. The finite element scheme possesses an $\mathcal{O}(h)$-order convergence rate for sufficiently regular data, and an $\mathcal{O}(h^s)$-order rate on any $s$-regular domain, $0<s\leqslant 1$, no matter what topology the domain has.
We propose a monotone, and consistent numerical scheme for the approximation of the Dirichlet problem for the normalized Infinity Laplacian, which could be related to the family of so--called two--scale methods. We show that this method is convergent, and prove rates of convergence. These rates depend not only on the regularity of the solution, but also on whether or not the right hand side vanishes. Some extensions to this approach, like obstacle problems and symmetric Finsler norms are also considered.
This paper investigate on numerical analysis on modified Single-layer approach to exterior Dirichlet problem of Laplace equation. We complete the convergence and error analysis of Petrov-Galerkin and Galerkin-Collocation methods with trigonometric basis for the induced modified Symm's integral equation of the first kind on analytic boundary. Besides, utilizing the composite trapezial quadrature formula and trigonometric interpolation to handle the singularity in modified logarithmic kernel, we establish the numerical procedure for implementation. On these numerical examples, we compare the eff…
Scaled lattice rules for integration on $\mathbb{R}^d$ achieving higher-order convergence with error analysis in terms of orthogonal projections onto periodic spaces
Nitsche-based FEM for the Laplace eigenvalue problem: spectral approximation and a posteriori error analysis
Indian Institute of Technology (IIT) Roorkee（印度理工学院鲁尔基分校）；Universidad del Bío-Bío（比奥比奥大学）；Universidad de Concepción（孔塞普西翁大学）；Universidad Técnica Federico Santa María（费德里科·圣玛丽亚技术大学）
In this paper we study finite element discretizations of a surface vector-Laplace eigenproblem. We consider two known classes of finite element methods, namely one based on a vector analogon of the Dziuk-Elliott surface finite element method and one based on the so-called trace finite element technique. A key ingredient in both classes of methods is a penalization method that is used to enforce tangentiality of the vector field in a weak sense. This penalization and the perturbations that arise from numerical approximation of the surface lead to essential nonconformities in the discretization …
Although fractional powers of non-negative operators have received much attention in recent years, there is still little known about their behavior if real-valued exponents are greater than one. In this article, we define and study the discrete fractional Laplace operator of arbitrary real-valued positive order. A series representation of the discrete fractional Laplace operator for positive non-integer powers is developed. Its convergence to a series representation of a known case of positive integer powers is proven as the power tends to the integer value. Furthermore, we show that the new r…
In this paper, a perfectly matched layer (PML) method is proposed to solve the time-domain electromagnetic scattering problems in 3D effectively. The PML problem is defined in a spherical layer and derived by using the Laplace transform and real coordinate stretching in the frequency domain. The well-posedness and the stability estimate of the PML problem are first proved based on the Laplace transform and the energy method. The exponential convergence of the PML method is then established in terms of the thickness of the layer and the PML absorbing parameter. As far as we know, this is the fi…
Finite element method to solve the spectral problem for arbitrary self-adjoint extensions of the Laplace-Beltrami operator on manifolds with a boundary
Univ. Carlos III de Madrid（马德里卡洛斯三世大学）；INFN, Sezione di Napoli（意大利国家核物理研究所那不勒斯分部）
