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LongTime Stability of Finite Element Approximations for Parabolic Equations with Memory
"... In this paper we derive the sharp longtime stability and error estimates of nite element approximations for parabolic integrodifferential equations. First, the exponential decay of the solution as t! 1 is studied, and then the semidiscrete and fully discrete approximations are considered using th ..."
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In this paper we derive the sharp longtime stability and error estimates of nite element approximations for parabolic integrodifferential equations. First, the exponential decay of the solution as t! 1 is studied, and then the semidiscrete and fully discrete approximations are considered using the RitzVolterra projection. Other related problems are studied as well. The main feature of our analysis is that the results are valid for both smooth and nonsmooth (weakly singular) kernels.
Preprint Number 09–25 H1SECOND ORDER CONVERGENT ESTIMATES FOR NON FICKIAN MODELS
"... Abstract: In this paper we study numerical methods for integrodifferential initial boundary value problems that arise, naturally, in many applications such as heat conduction in materials with memory, diffusion in polymers and diffusion in porous media. We propose finite difference methods to compu ..."
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Abstract: In this paper we study numerical methods for integrodifferential initial boundary value problems that arise, naturally, in many applications such as heat conduction in materials with memory, diffusion in polymers and diffusion in porous media. We propose finite difference methods to compute approximations for the continuous solutions of such problems. For those methods we analyze the stability and study the convergence. We prove a supraconvergent estimate. As such methods can be seen as lumped mass methods, our supraconvergent result is a superconvergent result in the context of finite element methods. Numerical results illustrating the theoretical results are included.