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Numer. Anal."],"published-print":{"date-parts":[[2018,1]]},"abstract":"<jats:p>The fractional Laplacian $\\Delta^{\\beta\/2}$ is the generator of the $\\beta$-stable L\u00e9vy process, which is the scaling limit of the L\u00e9vy flight. Due to the divergence of the second moment of the jump length of the L\u00e9vy flight, it may not be a suitable physical model in many practical applications. However, using a parameter $\\lambda$ to exponentially temper the isotropic power law measure of the jump length leads to the tempered L\u00e9vy flight, which has finite second moment. For a short time the tempered L\u00e9vy flight exhibits the dynamics of L\u00e9vy flight, while after a sufficiently long time it turns to normal diffusion. The generator of the tempered $\\beta$-stable L\u00e9vy process is the tempered fractional Laplacian $(\\Delta+\\lambda)^{\\beta\/2}$ [W. H. Deng et al., Multiscale Model. Simul., 16 (2018), pp. 125--149]. In the current work, we present new computational methods for the tempered fractional Laplacian equation, including the cases with the homogeneous and nonhomogeneous generalized Dirichlet type boundary conditions. We prove the well-posedness of the Galerkin weak formulation and provide convergence analysis of the single scaling B-spline and multiscale Riesz bases finite element methods. We propose a technique for efficiently generating the entries of the dense stiffness matrix and for solving the resulting algebraic equation by preconditioning. We also present several numerical experiments to verify the theoretical results.<\/jats:p>","DOI":"10.1137\/17m1151791","type":"journal-article","created":{"date-parts":[[2018,10,16]],"date-time":"2018-10-16T13:29:31Z","timestamp":1539696571000},"page":"3010-3039","source":"Crossref","is-referenced-by-count":45,"title":["A Riesz Basis Galerkin Method for the Tempered Fractional Laplacian"],"prefix":"10.1137","volume":"56","author":[{"given":"Zhijiang","family":"Zhang","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8573-012X","authenticated-orcid":true,"given":"Weihua","family":"Deng","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9713-7120","authenticated-orcid":true,"given":"George Em","family":"Karniadakis","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2018,10,11]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1016\/j.camwa.2017.05.026"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1137\/15M1033952"},{"key":"atypb3","volume-title":"Handbook of Mathematical Functions","author":"Abramowitz M.","year":"1965"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2009.10.027"},{"key":"atypb5","volume-title":"Finite Element Approximation for the Fractional Eigenvalue Problem, preprint, https:\/\/arxiv.org\/abs\/1603.00317","author":"Borthagaray J. 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