A mixed‐type Galerkin variational formulation and fast algorithms for variable‐coefficient fractional diffusion equations

Abstract

We consider the variable‐coefficient fractional diffusion equations with two‐sided fractional derivative. By introducing an intermediate variable, we propose a mixed‐type Galerkin variational formulation and prove the existence and uniqueness of the variational solution over H01(normalΩ)×H1−β2(normalΩ). On the basis of the formulation, we develop a mixed‐type finite element procedure on commonly used finite element spaces and derive the solvability of the finite element solution and the error bounds for the unknown and the intermediate variable. For the Toeplitz‐like linear system generated by discretization, we design a fast conjugate gradient normal residual method to reduce the storage from O(N2) to O(N) and the computing cost from O(N3) to O(NlogN). Numerical experiments are included to verify our theoretical findings. Copyright © 2017 John Wiley & Sons, Ltd.

Least‐squares mixed Galerkin formulation for variable‐coefficient fractional differential equations with D‐N boundary condition

“…In theoretical analysis, we derive optimal-order error estimates for the discrete solutions expressed in terms of the smoothness of the right-hand side only, which are apparently different from those estimates of the earlier Galerkin variational formulation. 15,18,20 The numerical experiments are conducted to verify our theoretical findings.…”

“…In the computation, we decouple the ( u , σ , p )−involved least‐squares variational formulation and establish a finite element procedure which can solve p , σ and u respectively. In theoretical analysis, we derive optimal‐order error estimates for the discrete solutions expressed in terms of the smoothness of the right‐hand side only, which are apparently different from those estimates of the earlier Galerkin variational formulation . The numerical experiments are conducted to verify our theoretical findings.…”

“…Remark 5.3. Compared with the earlier Galerkin finite elements, 15,18,20 the convergence orders for p, and u in Theorem 5.2 are optimal with respect to both the regularity of the solution and the index of the finite element space. Further, the error estimates in Theorem 5.2 are expressed in terms of the smoothness of the right-hand side f only, which may provide more accurate estimates in real applications.…”

“…This may increase the computation cost due to space‐reconstruction. Recently, Li et al proposed a mixed type Galerkin formulation for a similar fractional equation and proved the equivalence between the Galerkin formulation and the fractional differential equation under the higher regularity requirement of f ∈ H s (Ω) and u ∈ H 2 − β + s for This may rule out the weak singular part x 1 − β from the exact solution u . All these mentioned results were obtained under Dirichlet boundary conditions.…”

An expanded mixed finite element simulation for two-sided time-dependent fractional diffusion problem

“…In the series of works [22][23][24], Ervin and Roop presented a first rigorous analysis for the stationary fractional advection dispersion equation based on a variational formulation. Then the discontinuous Galerkin method [25], mixed finite element method [26][27][28][29][30], Petrov Galerkin method [31] and the least-squared mixed method are proposed [32] for stationary fractional diffusion equations, consecutively.…”