Finite element approximation of fractional order elliptic boundary value problems

Abstract

A finite element numerical method is investigated for fractional order elliptic boundary value problems with homogeneous Dirichlet type boundary conditions. It is pointed out that an appropriate stiffness matrix can be obtained by taking the prescribed fractional power of the stiffness matrix corresponding to the non-fractional elliptic operators.It is proved that this approach, which is also called the matrix transformation or matrix transfer method, delivers optimal rate of convergence in the L 2 -norm.

Törtrendű diffúzió modelljei és szimulációja

“…Attekintjük a különböző modellekhez tartozó numerikus módszereket. Az első hatékony eljárást a [12] munkában közölték, amelynek alapgondolata, hogy el kell tolni a véges differenciában szereplő együtthatókat valamilyen p ∈ Z + paraméterrel: [12]és [17], a másodikéhoz a [14]és [15] munkákat idézzük.…”

Numerical Solution of Time-Dependent Problems with Fractional Power Elliptic Operator

“…The use of finite element approximations for fractional power elliptic operators is discussed in detail, for instance, in the works Acosta and Borthagaray (2017); Szekeres and Izsák (2016).…”

“…For approximation in space, we can apply finite volume or finite element methods oriented to using arbitrary domains and irregular computational grids (Knabner and Angermann (2003); Quarteroni and Valli (1994)). After this, we formulate the corresponding Cauchy problem with a fractional power of a self-adjoint positive definite discrete elliptic operator (see Bonito and Pasciak (2015); Szekeres and Izsák (2016)) -a fractional power of a symmetric positive definite matrix (Higham (2008)).…”

Periodic boundary value problems for two classes of nonlinear fractional differential equations

“…In the past few decades, boundary value problems of fractional order involving a variety of boundary conditions have been studied by several researchers. We refer the readers to [9][10][11][12][13][14][15] and the references cited therein. Moreover,the existence of solutions to the fractional differential equations with anti-periodic boundary value conditions has been studied by many authors (see [16][17][18][19][20][21]).…”