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Error estimates for a semidiscrete finite element method for fractional order parabolic equations. (English) Zbl 1268.65126
The authors consider the initial boundary value problem for a homogeneous time-fractional diffusion equation with an initial condition $$v(x)$$ and a homogeneous Dirichlet boundary condition in a bounded convex polygonal domain $$\Omega$$. Two semidiscrete approximation schemes, i.e. the Galerkin finite element method (FEM) and lumped mass Galerkin FEM, using piecewise linear functions are studied. Almost optimal error estimates with respect to the data regularity, including the cases of smooth and nonsmooth initial data, i.e., $$v \in H^2(\Omega)\cap H^1_0(\Omega)$$ and $$v \in L_2(\Omega)$$ are established. 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. Some numerical results that give insight into the reliability of the theoretical study are presented.

##### MSC:
 65M15 Error bounds for initial value and initial-boundary value problems involving PDEs 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 65M20 Method of lines for initial value and initial-boundary value problems involving PDEs 35K05 Heat equation 35R11 Fractional partial differential equations
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