Uniform exponential long time decay for the space semi-discretization of a locally damped wave equation via an artificial numerical viscosity

Abstract

We consider the finite-difference space semi-discretization of a locally damped wave equation, the damping being supported in a suitable subset of the domain under consideration, so that the energy of solutions of the damped wave equation decays exponentially to zero as time goes to infinity. The decay rate of the semi-discrete systems turns out to depend on the mesh size h of the discretization and tends to zero as h goes to zero. We prove that adding a suitable vanishing numerical viscosity term leads to a uniform (with respect to the mesh size) exponential decay of the energy of solutions. This numerical viscosity term damps out the high frequency numerical spurious oscillations while the convergence of the scheme towards the original damped wave equation is kept. We discuss this problem in 1D and 2D in the interval and the square respectively. Our method of proof relies on discrete multiplier techniques. (1991): 65M06 * Current address: Mathematics Subject Classification

Observability properties of a semi-discrete 1d wave equation derived from a mixed finite element method on nonuniform meshes

“…Similarly as in the continuous setting, this property is equivalent to the uniform observability inequality (1.12) for solutions of the conservative system (1.7) (see for instance [27]). Therefore Theorem 1.2 leads to the following result:…”

“…In particular, if there exists a constant M 2 > 0 such that for all n, 27) then, for all n and k, we get…”

“…The proof of Theorem 4.1, which can be adapted in a standard way from [16] or [27], is left to the reader.…”

“…A new approach was proposed recently in [25] based on wavelet filtering techniques. Let us also mention the results [10,[26][27][28] that amounts to adding an extra term in (1.12) which is non-negligible only for the high frequencies. A last possible cure was proposed in [1,15] and later analyzed in [5]: a 1d semi-discrete scheme derived from a mixed finite element method was proposed, which has the property that the group velocity of the waves is bounded from below.…”

Energetic BEM for the numerical analysis of 2D Dirichlet damped wave propagation exterior problems

“…Hence, damping is whether an unavoidable presence in physical reality or a desired characteristic in design. The use of advanced numerical techniques to solve the related PDEs, such as FEM and the finite difference methods (FDM) is well established and it is standard in this framework , even if the research of a numerical method that could reproduce the expected damping decay is an actual argument in literature (see [14][15][16] and their references). On the other hand, in the context of BEMs the analysis of dissipation through damped wave equation rewritten as a BIE is a relatively new topic, because it has been scarcely investigated until now.…”