Journal
SIAM JOURNAL ON NUMERICAL ANALYSIS
Volume 51, Issue 1, Pages 445-466Publisher
SIAM PUBLICATIONS
DOI: 10.1137/120873984
Keywords
finite element method; fractional diffusion; optimal error estimates; semidiscrete Gelerkin method; lumped mass method
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Funding
- US NSF [DMS-1016525]
- King Abdullah University of Science and Technology (KAUST) [KUS-C1-016-04]
- Division Of Mathematical Sciences
- Direct For Mathematical & Physical Scien [1016525] Funding Source: National Science Foundation
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We consider the initial boundary value problem for a homogeneous time-fractional diffusion equation with an initial condition nu(x) and a homogeneous Dirichlet boundary condition in a bounded convex polygonal domain Omega. We study two semidiscrete approximation schemes, i. e., the Galerkin finite element method (FEM) and lumped mass Galerkin FEM, using piecewise linear functions. We establish almost optimal with respect to the data regularity error estimates, including the cases of smooth and nonsmooth initial data, i. e., nu is an element of H-2(Omega) boolean AND H-0(1)(Omega) and nu is an element of L-2(Omega). For the lumped mass method, the optimal L-2-norm error estimate is valid only under an additional assumption on the mesh, which in two dimensions is known to be satisfied for symmetric meshes. Finally, we present some numerical results that give insight into the reliability of the theoretical study.
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