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）
Lehel Banjai, Jens M. Melenk, Ricardo H. Nochetto, Enrique Otarola, Abner J. Salgado, Christoph Schwab
We design and analyze several Finite Element Methods (FEMs) applied to the Caffarelli-Silvestre extension that localizes the fractional powers of symmetric, coercive, linear elliptic operators in bounded domains with Dirichlet boundary conditions. We consider open, bounded, polytopal but not necessarily convex domains $Ω\subset \mathbb{R}^d$ with $d=1,2$. For the solution to the extension problem, we establish analytic regularity with respect to the extended variable $y\in (0,\infty)$. We prove that the solution belongs to countably normed, power-exponentially weighted Bochner spaces of analytic functions with respect to $y$, taking values in corner-weighted Kondat'ev type Sobolev spaces in $Ω$. In $Ω\subset \mathbb{R}^d$, we discretize with continuous, piecewise linear, Lagrangian FEM ($P_1$-FEM) with mesh refinement near corners, and prove that first order convergence rate is attained for compatible data $f\in \mathbb{H}^{1-s}(Ω)$.
We also prove that tensorization of a $P_1$-FEM in $Ω$ with a suitable $hp$-FEM in the extended variable achieves log-linear complexity with respect to $\mathcal{N}_Ω$, the number of degrees of freedom in the domain $Ω$. In addition, we propose a novel, sparse tensor product FEM based on a multilevel $P_1$-FEM in $Ω$ and on a $P_1$-FEM on radical-geometric meshes in the extended variable. We prove that this approach also achieves log-linear complexity with respect to $\mathcal{N}_Ω$. Finally, under the stronger assumption that the data is analytic in $\overlineΩ$, and without compatibility at $\partial Ω$, we establish exponential rates of convergence of $hp$-FEM for spectral, fractional diffusion operators. We also report numerical experiments for model problems which confirm the theoretical results. We indicate several extensions and generalizations of the proposed methods.
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.
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）
Homogenization in terms of multiscale limits transforms a multiscale problem with $n+1$ asymptotically separated microscales posed on a physical domain $D \subset \mathbb{R}^d$ into a one-scale problem posed on a product domain of dimension $(n+1)d$ by introducing $n$ so-called "fast variables". This procedure allows to convert $n+1$ scales in $d$ physical dimensions into a single-scale structure in $(n+1)d$ dimensions. We prove here that both the original, physical multiscale problem and the corresponding high-dimensional, one-scale limiting problem can be efficiently treated numerically with…
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…
We consider the one-constant Landau - de Gennes model for nematic liquid crystals. The order parameter is a traceless tensor field $\mathbf{Q}$, which is constrained to be uniaxial: $\mathbf{Q} = s (\mathbf{n}\otimes\mathbf{n} - d^{-1} \mathbf{I})$ where $\mathbf{n}$ is a director field, $s\in\mathbb{R}$ is the degree of orientation, and $d\ge2$ is the dimension. Building on similarities with the one-constant Ericksen energy, we propose a structure-preserving finite element method for the computation of equilibrium configurations. We prove stability and consistency of the method without regula…
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）
The use of fractional differential equations is a key tool in modeling non-local phenomena. Often, an efficient scheme for solving a linear system involving the discretization of a fractional operator is evaluating the matrix function $x = \mathcal A^{-α} c$, where $\mathcal A$ is a discretization of the classical Laplacian, and $α$ a fractional exponent between $0$ and $1$. In this work, we derive an exponential sum approximation for $f(z) =z^{-α}$ that is accurate over $[1, \infty)$ and allows to efficiently approximate the action of bounded and unbounded operators of this kind on tensors st…
Mathematical and numerical study of symmetry and positivity of the tensor-valued spring constant defined from P1-FEM for two- and three-dimensional linear elasticity
Graduate School of Natural Science and Technology, Kanazawa University（金泽大学自然科学技术研究科）；Faculty of Mathematics and Physics, Kanazawa University（金泽大学数理学院）
We introduce the tensor numerical method for solution of the $d$-dimensional optimal control problems with fractional Laplacian type operators in constraints discretized on large $n^{\otimes d}$ tensor-product Cartesian grids. The approach is based on the rank-structured approximation of the matrix valued functions of the corresponding fractional finite difference Laplacian. We solve the equation for the control function, where the system matrix includes the sum of the fractional $d$-dimensional Laplacian and its inverse. The matrix valued functions of discrete Laplace operator on a tensor gri…
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…
We introduce the tensor numerical method for solving optimal control problems that are constrained by fractional 2D and 3D elliptic operators with variable coefficients. We solve the governing equation for the control function which includes a sum of the fractional operator and its inverse, both discretized over large 3D $n\times n \times n$ spacial grids. Using the diagonalization of the arising matrix valued functions in the eigenbasis of the 1D Sturm-Liouville operators, we construct the rank-structured tensor approximation with controllable precision for the discretized fractional elliptic…
We propose a multi-index algorithm for the Monte Carlo (MC) discretization of a linear, elliptic PDE with affine-parametric input. We prove an error vs. work analysis which allows a multi-level finite-element approximation in the physical domain, and apply the multi-index analysis with isotropic, unstructured mesh refinement in the physical domain for the solution of the forward problem, for the approximation of the random field, and for the Monte-Carlo quadrature error. Our approach allows Lipschitz domains and mesh hierarchies more general than tensor grids. The improvement in complexity ove…
A large class of scalar-tensor theories of gravity exhibit a screening mechanism that dynamically suppresses fifth forces in the Solar system and local laboratory experiments. Technically, at the scalar field equation level, this usually translates into nonlinearities which strongly limit the scope of analytical approaches. This article presents $femtoscope$ $-$ a Python numerical tool based on the Finite Element Method (FEM) and Newton method for solving Klein-Gordon-like equations that arise in particular in the symmetron or chameleon models. Regarding the latter, the scalar field behavior i…
