A fractional model for anomalous diffusion with increased variability: Analysis, algorithms and applications to interface problems
Copyright policy )A fractional model for anomalous diffusion with increased variability: Analysis, algorithms and applications to interface problems
AbstractFractional equations have become the model of choice in several applications where heterogeneities at the microstructure result in anomalous diffusive behavior at the macroscale. In this work we introduce a new fractional operator characterized by a doubly‐variable fractional order and possibly truncated interactions. Under certain conditions on the model parameters and on the regularity of the fractional order we show that the corresponding Poisson problem is well‐posed. We also introduce a finite element discretization and describe an efficient implementation of the finite‐element matrix assembly in the case of piecewise constant fractional order. Through several numerical tests, we illustrate the improved descriptive power of this new operator across media interfaces. Furthermore, we present one‐dimensional and two‐dimensional h‐convergence results that show that the variable‐order model has the same convergence behavior as the constant‐order model.
- Sandia National Laboratories United States
Integral operators, subsurface diffusion, Dirichlet forms, Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation, Applications of operator theory to differential and integral equations, Smoothness and regularity of solutions to PDEs, interface problems, Existence problems for PDEs: global existence, local existence, non-existence, Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness, Numerical Analysis (math.NA), Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs, Stability and convergence of numerical methods for boundary value problems involving PDEs, Fractional partial differential equations, Fractional derivatives and integrals, anomalous diffusion, Miscellaneous topics in partial differential equations, FOS: Mathematics, Mathematics - Numerical Analysis, variable-order fractional operators
Integral operators, subsurface diffusion, Dirichlet forms, Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation, Applications of operator theory to differential and integral equations, Smoothness and regularity of solutions to PDEs, interface problems, Existence problems for PDEs: global existence, local existence, non-existence, Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness, Numerical Analysis (math.NA), Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs, Stability and convergence of numerical methods for boundary value problems involving PDEs, Fractional partial differential equations, Fractional derivatives and integrals, anomalous diffusion, Miscellaneous topics in partial differential equations, FOS: Mathematics, Mathematics - Numerical Analysis, variable-order fractional operators
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