For the spectral fractional diffusion operator of order $2s\in (0,2)$ in bounded, curvilinear polygonal domains $Ω$ we prove exponential convergence of two classes of $hp$ discretizations under the assumption of analytic data, without any boundary compatibility, in the natural fractional Sobolev norm $\mathbb{H}^s(Ω)$. The first $hp$ discretization is based on writing the solution as a co-normal derivative of a $2+1$-dimensional local, linear elliptic boundary value problem, to which an $hp$-FE discretization is applied. A diagonalization in the extended variable reduces the numerical approximation of the inverse of the spectral fractional diffusion operator to the numerical approximation of a system of local, decoupled, second order reaction-diffusion equations in $Ω$. Leveraging results on robust exponential convergence of $hp$-FEM for second order, linear reaction diffusion boundary value problems in $Ω$, exponential convergence rates for solutions $u\in \mathbb{H}^s(Ω)$ of $\mathcal{L}^s u = f$ follow. Key ingredient in this $hp$-FEM are boundary fitted meshes with geometric mesh refinement towards $\partialΩ$.
The second discretization is based on exponentially convergent sinc quadrature approximations of the Balakrishnan integral representation of $\mathcal{L}^{-s}$, combined with $hp$-FE discretizations of a decoupled system of local, linear, singularly perturbed reaction-diffusion equations in $Ω$. The present analysis for either approach extends to polygonal subsets $\widetilde{\mathcal{M}}$ of analytic, compact $2$-manifolds $\mathcal{M}$. Numerical experiments for model problems in nonconvex polygonal domains and with incompatible data confirm the theoretical results.
Exponentially small bounds on Kolmogoroff $n$-widths of solutions sets for spectral fractional diffusion in polygons are deduced.
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）
$H^1$-conforming Galerkin methods on polygonal meshes such as VEM, BEM-FEM and Trefftz-FEM employ local finite element functions that are implicitly defined as solutions of Poisson problems having polynomial source and boundary data. Recently, such methods have been extended to allow for mesh cells that are curvilinear polygons. Such extensions present new challenges for determining suitable quadratures. We describe an approach for integrating products of these implicitly defined functions, as well as products of their gradients, that reduces integrals on cells to integrals along their boundar…
In this paper, we study diagonal dominance of the stiffness matrix resulted from the piecewise linear finite element discretisation of the integral fractional Laplacian under global homogeneous Dirichlet boundary condition in one spatial dimension. We first derive the exact form of this matrix in the frequency space which is extendable to multi-dimensional rectangular elements. Then we give the complete answer when the stiffness matrix can be strictly diagonally dominant. As one application, we apply this notion to the construction of maximum principle preserving schemes for the fractional-in-…
In this paper, we develop a linearized fractional Crank-Nicolson-Galerkin FEM for Kirchhoff type quasilinear time-fractional integro-differential equation $\left(\mathcal{D}^α\right)$. In general, the solutions to the time-fractional problems exhibit a weak singularity at time $t=0$. This singular behavior of the solutions is taken into account while deriving the convergence estimates of the developed numerical scheme. We prove that the proposed numerical scheme has an accuracy rate of $O(M^{-1}+N^{-2})$ in $L^{\infty}(0,T;L^{2}(Ω))$ as well as in $L^{\infty}(0,T;H^{1}_{0}(Ω))$, where $M$ and …
We derive and analyze a fully computable discrete scheme for fractional partial differential equations posed on the full space $\mathbb{R}^d$ . Based on a reformulation using the well-known Caffarelli-Silvestre extension, we study a modified variational formulation to obtain well-posedness of the discrete problem. Our scheme is obtained by combining a diagonalization procedure with a reformulation using boundary integral equations and a coupling of finite elements and boundary elements. For our discrete method we present a-priori estimates as well as numerical examples.
A semidiscrete Galerkin finite element method applied to time-fractional diffusion equations with time-space dependent diffusivity on bounded convex spatial domains will be studied. The main focus is on achieving optimal error results with respect to both the convergence order of the approximate solution and the regularity of the initial data. By using novel energy arguments, for each fixed time $t$, optimal error bounds in the spatial $L^2$- and $H^1$-norms are derived for both cases: smooth and nonsmooth initial data.
We derive exact form of the piecewise-linear finite element stiffness matrix on general non-uniform meshes for the integral fractional Laplacian operator in one dimension, where the derivation is accomplished in the Fourier transformed space. With such an exact formulation at our disposal, we are able to numerically study some intrinsic properties of the fractional stiffness matrix on some commonly used non-uniform meshes (e.g., the graded mesh), in particular, to examine their seamless transition to those of the usual Laplacian.
Towards an Efficient Finite Element Method for the Integral Fractional Laplacian on Polygonal Domains
Division of Applied Mathematics, Brown University（布朗大学应用数学系）；Computer Science and Mathematics Division, Oak Ridge National Laboratory（橡树岭国家实验室计算机科学与数学部）
This paper focuses on numerical approximation for fractional powers of elliptic operators on $2$-d manifolds. Firstly, parametric finite element method is employed to discretize the original problem. We then approximate fractional powers of the discrete elliptic operator by the product of rational functions, each of which is a diagonal Padé approximant for corresponding power function. Rigorous error analysis is carried out and sharp error bounds are presented which show that the scheme is robust for $α\rightarrow 0^+$ and $α\rightarrow 1^-$. The cost of the proposed algorithm is solving some …
Non-uniform $α$-Robust Alikhanov Mixed FEM with Optimal Convergence for the Time-Fractional Allen--Cahn Equation
Indian Institute of Technology Gandhinagar（印度理工学院甘地纳加尔分校）；Sultan Qaboos University（苏丹卡布斯大学）；TU Wien, Institut für Analysis und Scientific Computing（维也纳工业大学分析与科学计算研究所）
Space-Time Petrov-Galerkin FEM for Fractional Diffusion Problems
Central South University（中南大学）；Texas A&M University（德克萨斯农工大学）；University College London（伦敦大学学院）；Bulgarian Academy of Sciences（保加利亚科学院）；Columbia University（哥伦比亚大学）
Heriot-Watt University（赫瑞-瓦特大学）；Technische Universität Wien（维也纳工业大学）；University of Maryland（马里兰大学）；Universidad Técnica Federico Santa María（费德里科·圣塔玛利亚技术大学）；University of Tennessee（田纳西大学）；ETH Zürich（苏黎世联邦理工学院）
In this paper, we study a time-fractional initial-boundary value problem of Kirchhoff type involving memory term for non-homogeneous materials. The energy argument is applied to derive the a priori bounds on the solution of the considered problem. Consequently, we prove the existence and uniqueness of the weak solution to the problem under consideration. We keep the time variable continuous and discretize the space domain using a conforming FEM to obtain the semi discrete formulation of the problem. The semi discrete error analysis is carried out by modifying the standard Ritz-Volterra project…
A PDE approach to fractional diffusion: a space-fractional wave equation
Maxwell Institute for Mathematical Sciences, School of Mathematical & Computer Sciences, Heriot-Watt University（赫瑞-瓦特大学数学与计算机科学学院）；Departamento de Matemática, Universidad Técnica Federico Santa María（弗雷德里科·圣塔玛丽亚理工大学数学系）
Arbitrary order 2D virtual elements for polygonal meshes: Part II, inelastic problem
University of Rome Tor Vergata（罗马托尔维加塔大学）；University of Milano（米兰大学）；University of Cassino and Southern Lazio（卡西诺与南拉齐奥大学）；University of Milan Bicocca（米兰比可卡大学）
We consider fractional partial differential equations posed on the full space $\R^d$. Using the well-known Caffarelli-Silvestre extension to $\R^d \times \R^+$ as equivalent definition, we derive existence and uniqueness of weak solutions. We show that solutions to a truncated extension problem on $\R^d \times (0,\YY)$ converge to the solution of the original problem as $\YY \rightarrow \infty$. Moreover, we also provide an algebraic rate of decay and derive weighted analytic-type regularity estimates for solutions to the truncated problem. These results pave the way for a rigorous analysis of…
This study presents the analytical and finite element formulation of a geometrically nonlinear and fractional-order nonlocal model of an Euler-Bernoulli beam. The finite nonlocal strains in the Euler-Bernoulli beam are obtained from a frame-invariant and dimensionally consistent fractional-order (nonlocal) continuum formulation. The finite fractional strain theory provides a positive definite formulation that results in a mathematically well-posed formulation which is consistent across loading and boundary conditions. The governing equations and the corresponding boundary conditions of the geo…
We present the analytical formulation and the finite element solution of a fractional-order nonlocal continuum model of a Euler-Bernoulli beam. Employing consistent definitions for the fractional-order kinematic relations, the governing equations and the associated boundary conditions are derived based on variational principles. Remarkably, the fractional-order nonlocal model gives rise to a self-adjoint and positive-definite system accepting a unique solution. Further, owing to the difficulty in obtaining analytical solutions to this boundary value problem, a finite element model for the frac…
Veamy: an extensible object-oriented C++ library for the virtual element method
Universidad de Chile（智利大学）；Università di Milano-Bicocca（米兰比可卡大学）；Istituto di Matematica Applicata e Tecnologie Informatiche del CNR（意大利国家研究委员会应用数学与信息技术研究所）
This article concerns the weak Galerkin mixed finite element method (WG-MFEM) for second order elliptic equations on 2D domains with curved boundary. The Neumann boundary condition is considered since it becomes the essential boundary condition in this case. It is well-known that the discrepancy between the curved physical domain and the polygonal approximation domain leads to a loss of accuracy for discretization with polynomial order $α>1$. The purpose of this paper is two-fold. First, we present a detailed error analysis of the original WG-MFEM for solving problems on curved domains, which …
