Nonnegativity Preservation

Abstract

It is known that for time-fractional diffusion, the standard weak maximum principle holds. Hence the solution is nonnegative for nonnegative initial data and the source $$f\equiv 0$$ . It is natural to ask whether this property is inherited by certain spatially semidiscrete and fully discrete piecewise linear fems, including the standard Galerkin method, lumped mass method, and finite volume element method. In this chapter, we discuss the nonnegativity preservation property for a class of piecewise linear finite element approximations for the following linear homogeneous time-fractional problem.