We present and study a novel numerical algorithm to approximate the action of T β := L −β where L is a symmetric and positive definite unbounded operator on a Hilbert space H 0 . The numerical method is based on a representation formula for T −β in terms of Bochner integrals involving (I + t 2 L) −1 for t ∈ (0, ∞).To develop an approximation to T β , we introduce a finite element approximation L h to L and base our approximation to T β on T β h := L −β h . The direct evaluation of T β h is extremely expensive as it involves expansion in the basis of eigenfunctions for L h . The above mentioned representation formula holds for T −β h and we propose three quadrature approximations denoted generically by Q β h . The two results of this paper bound the errors in the H 0 inner product of T β − T β h π h and T β h − Q β h where π h is the H 0 orthogonal projection into the finite element space. We note that the evaluation of Q β h involves application of (I + (t i ) 2 L h ) −1 with t i being either a quadrature point or its inverse. Efficient solution algorithms for these problems are available and the problems at different quadrature points can be straightforwardly solved in parallel. Numerical experiments illustrating the theoretical estimates are provided for both the quadrature error T β h − Q β h and the finite element error T β − T β h π h .
Abstract. We analyze an adaptive discontinuous finite element method (ADFEM) for symmetric second order linear elliptic operators. The method is formulated on nonconforming meshes made of simplices or quadrilaterals, with any polynomial degree and in any dimension ≥ 2. We prove that the ADFEM is a contraction for the sum of the energy error and the scaled error estimator, between two consecutive adaptive loops. We design a refinement procedure that maintains the level of nonconformity uniformly bounded, and prove that the approximation classes using continuous and discontinuous finite elements are equivalent. The geometric decay and the equivalence of classes are instrumental to derive optimal cardinality of ADFEM. We show that ADFEM (and AFEM on nonconforming meshes) yields a decay rate of energy error plus oscillation in terms of number of degrees of freedom as dictated by the best approximation for this combined nonlinear quantity.
We present three schemes for the numerical approximation of fractional diffusion, which build on different definitions of such a non-local process. The first method is a PDE approach that applies to the spectral definition and exploits the extension to one higher dimension. The second method is the integral formulation and deals with singular non-integrable kernels. The third method is a discretization of the Dunford-Taylor formula. We discuss pros and cons of each method, error estimates, and document their performance with a few numerical experiments.Date: Draft version of July 7, 2017.
We propose a new nonconforming finite element algorithm to approximate the solution to the elliptic problem involving the fractional Laplacian. We first derive an integral representation of the bilinear form corresponding to the variational problem. The numerical approximation of the action of the corresponding stiffness matrix consists of three steps: (i) apply a sinc quadrature scheme to approximate the integral representation by a finite sum where each term involves the solution of an elliptic partial differential equation defined on the entire space, (ii) truncate each elliptic problem to a bounded domain, (iii) use the finite element method for the space approximation on each truncated domain. The consistency error analysis for the three steps is discussed together with the numerical implementation of the entire algorithm. The results of computations are given illustrating the error behavior in terms of the mesh size of the physical domain, the domain truncation parameter and the quadrature spacing parameter.Date: December 20, 2018. 1 arXiv:1707.04290v2 [math.NA] 18 Dec 2018where (·, ·) denotes the inner product on L 2 (R d ) (see, also [3]). We note that for t > 0, (I − t 2 ∆) −1 is a bounded map of L 2 (R d ) into H 2 (R d ) so that the integrand above is well defined for η, θ ∈ L 2 (R d ). In Theorem 4.1, we show that for η ∈ H r (R d ) and θ ∈ H s−r (R d ), the formula (13) holds and the right hand side integral converges absolutely. It follows that the bilinear form a(·, ·) is given by (14) a(η, θ) = c s ∞ 0 t 2−2s ((−∆)(I − t 2 ∆) −1 η, θ) D dt t , for all η, θ ∈ H s (D).
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