A posteriori error bounds for pseudo parabolic problems using C0 semigroups

Martin Ossadnik (FernUniversität in Hagen), Torsten Linß

We consider the second-order pseudo parabolic equation of finding u:[0,T]H01(Ω), ΩRd, such that Ltu(t)+Mu(t)=F(t)in (0,T] with two second order, elliptic operators L,M:H01(Ω)H1(Ω) and a source function F:[0,T]H1(Ω). Furthermore an initial condition u(0)=u0,u0H01(Ω), is given. We will assume, that the operator M is bounded and that L is both bounded and coercive.

A computable a posteriori error bound in a weighted H1-norm for a full discretisation using the backward differential formula of order two (BDF-2 method) in time and P2-elements in space is derived. To do so, we leverage the C0 semigroup, generated by the operator L1M. Furthermore we adapt elliptic reconstructions introduced by C. Makridakis and R. N. Nochetto to pseudo parabolic problems.
Given some numerical results we analyze the estimate’s order, efficiency and components. Last but not least, we show that we can apply our a posteriori error bound to other time discretisations like the backward Euler and Crank Nicolson method.

References

  1. Ch. Makridakis and R. H. Nochetto. Elliptic reconstruction and a posteriori error estimates for parabolic problems. SIAM J. Numer. Anal., 41(4):1585–1594, 2003.

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