Mass Lumping for Constant-Density Acoustics
Symes, William W.
Conforming Galerkin discretization of the constant-density acoustic wave equation provides optimal order convergence even in the presence of very rough coefficients, provided that the time dependence of the data (right-hand side) is minimally smooth. Such discretizations avoid the well-known first-order error ("stairstep diffraction") phenomenon produced by standard finite difference methods. On the other hand, Galerkin methods in themselves are inefficient for high frequency wave simulation, due to the implicit nature of the time step system. Mass lumping renders the time step explicit, and provides an avenue for efficient time-stepping of time-dependent problems with conforming finite element spatial discretization. Typical justifications for mass lumping use quadrature error estimates which do not hold for nonsmooth coefficients. In this paper, we show that the mass-lumped semidiscrete system for the constant-density acoustic wave equation with rectangular multilinear elements exhibits optimal order convergence even when the coefficient (bulk modulus) is merely bounded and measurable, provided that the right-hand side possesses some smoothness in time. We illustrate the theory with numerical examples involving discontinuous, non-grid-aligned bulk moduli, in which the coefficient averaging implicit in mass lumping eliminates the stairstep diffraction effect.
Citable link to this pagehttps://hdl.handle.net/1911/102111
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