Abstract
We propose a cheaper version ofa posteriorierror estimator from Goryninaet al.(Numer. Anal.(2017)) for the linear second-order wave equation discretized by the Newmark scheme in time and by the finite element method in space. The new estimator preserves all the properties of the previous one (reliability, optimality on smooth solutions and quasi-uniform meshes) but no longer requires an extra computation of the Laplacian of the discrete solution on each time step.
Highlights
In this paper, we are interested in a posteriori time-space error estimates for finite element discretizations of the wave equation
For instance, in [9, 12] for the case of implicit Euler discretization in time, in [13] for the case of the second order discretization in time by Cosine scheme, and in [15] for a particular variant of the Newmark scheme β = 1/4, γ = 1/2. In both [13, 15], the error is measured in a physically natural norm: H1 in space, L∞ in time. Another common feature of these two papers is that the time error estimators proposed there contain the Laplacian of the discrete solution which should be computed via an auxiliary finite element problem at each time step
We propose an alternative time error estimator for the particular Newmark scheme considered in [15] that avoids these additional computations
Summary
We are interested in a posteriori time-space error estimates for finite element discretizations of the wave equation Such lestimates were designed, for instance, in [9, 12] for the case of implicit Euler discretization in time, in [13] for the case of the second order discretization in time by Cosine (or, equivalently, Newmark) scheme, and in [15] for a particular variant of the Newmark scheme β = 1/4, γ = 1/2 (the advantage of the approach from [15] being its suitability for non constant time steps while the estimator from [13] is restricted to uniform meshes in time). Numerical experiments demonstrate that 3-point and 5-point error estimators produce very similar results in the majority of test cases Both turn out to be of optimal order in space and time, even in situations not accessible to the current theory (non quasi-uniform meshes, not constant time steps).
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