Parallel solvers for fractional power diffusion problems

2017 ◽  
Vol 29 (24) ◽  
pp. e4216 ◽  
Author(s):  
Raimondas Čiegis ◽  
Vadimas Starikovičius ◽  
Svetozar Margenov ◽  
Rima Kriauzienė
Keyword(s):  
Fractional Power ◽  
Parallel Solvers ◽  
Power Diffusion ◽  
Diffusion Problems

Scalability analysis of different parallel solvers for 3D fractional power diffusion problems

2019 ◽  
Vol 31 (19) ◽  
Author(s):  
Raimondas Čiegis ◽  
Vadimas Starikovičius ◽  
Svetozar Margenov ◽  
Rima Kriauzienė
Keyword(s):  
Fractional Power ◽  
Parallel Solvers ◽  
Scalability Analysis ◽  
Power Diffusion ◽  
Diffusion Problems

scholarly journals Reduced basis approximations of the solutions to spectral fractional diffusion problems

2020 ◽  
Vol 28 (3) ◽  
pp. 147-160
Author(s):  
Andrea Bonito ◽  
Diane Guignard ◽  
Ashley R. Zhang
Keyword(s):  
Computational Cost ◽  
Fractional Power ◽  
Reaction Diffusion ◽  
Fractional Diffusion ◽  
Diffusion Problem ◽  
Quadrature Point ◽  
Reduced Basis ◽  
Fast Evaluation ◽  
Bottle Neck ◽  
Diffusion Problems

AbstractWe consider the numerical approximation of the spectral fractional diffusion problem based on the so called Balakrishnan representation. The latter consists of an improper integral approximated via quadratures. At each quadrature point, a reaction–diffusion problem must be approximated and is the method bottle neck. In this work, we propose to reduce the computational cost using a reduced basis strategy allowing for a fast evaluation of the reaction–diffusion problems. The reduced basis does not depend on the fractional power s for 0 < smin ⩽ s ⩽ smax < 1. It is built offline once for all and used online irrespectively of the fractional power. We analyze the reduced basis strategy and show its exponential convergence. The analytical results are illustrated with insightful numerical experiments.

Sharp estimates for homogeneous semigroups in homogeneous spaces. Applications to PDEs and fractional diffusion in ℝN

2020 ◽  
pp. 2050070
Author(s):  
Jan W. Cholewa ◽  
Anibal Rodriguez-Bernal
Keyword(s):  
Stokes Equations ◽  
Homogeneous Spaces ◽  
Fractional Power ◽  
Fractional Diffusion ◽  
Morrey Spaces ◽  
Higher Order ◽  
Parabolic Problems ◽  
Evolution Problems ◽  
Sharp Estimates ◽  
Diffusion Problems

In this paper, we analyze evolution problems associated to homogenous operators. We show that they have an homogenous associated semigroup of solutions that must satisfy some sharp estimates when acting on homogenous spaces and on the associated fractional power spaces. These sharp estimates are determined by the homogeneity alone. We also consider fractional diffusion problems and Schrödinger type problems as well. We apply these general results to broad classes of PDE problems including heat or higher order parabolic problems and the associated fractional and Schrödinger problems or Stokes equations. These equations are considered in Lebesgue or Morrey spaces.

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