Preasymptotic Error Analysis of High Order Interior Penalty Discontinuous Galerkin Methods for the Helmholtz Equation with High Wave Number

Abstract

A preasymptotic error analysis of the interior penalty discontinuous Galerkin (IPDG) method of high order for Helmholtz equation with the first order absorbing boundary condition in two and three dimensions is proposed. We derive the H 1 -and L 2 -error estimates with explicit dependence on the wave number k. In particular, it is shown that if k(kh) 2 p is sufficiently small, then the pollution errors of IPDG method in H 1 -norm are bounded by O(k(kh) 2 p ), which coincides with the phase error of the finite element method obtained by existent dispersion analyses on Cartesian grids, where h is the mesh size, p is the order of the approximation space and is fixed. Numerical tests are provided to verify the theoretical findings and to illustrate great capability of the symmetric IPDG method in reducing the pollution effect.

A sharp relative-error bound for the Helmholtz h-FEM at high frequency

“…As discussed above, these are the first-ever frequency-explicit relative-error bounds on the Helmholtz h-FEM in 2 or 3 dimensions. We recall the interest (highlighted at the end of the previous subsection) from [14,18,[32][33][34][35]44,51,76,[100][101][102][103][104] in proving such bounds. An additional novelty of Theorem 4.1 is that it applies to the variable-coefficient Helmholtz equation, and all the constants in the relative-error bound are explicit, not only in k and h, but also in the coefficients A and n. The only other coefficient-explicit, preasymptotic FEM error bound on the variable-coefficient Helmholtz equation in the literature appears in [87,Theorem 2.39], where the bound (1.3) is proved for the interior impedance problem when h 2 p k 2 p+1 is sufficiently small and A and n are nontrapping.…”

“…We highlight that, while [32,51,76] all prove results of the form (1.3), all the numerical experiments in these papers consider the relative error (either in the H 1 norm [32,76], or the weighted H 1 norm (3.2) [51]), illustrating that relative error is indeed the quantity of interest in practice. An analogous situation is encountered in the preasymptotic error analyses of other Helmholtz FEMs in [14,18,[33][34][35]44,[101][102][103]: all these papers prove bounds on the error in terms of the data, as in (1.3), but all the numerical experiments in these papers concerning the error consider the relative error.…”

Decompositions of high-frequency Helmholtz solutions via functional calculus, and application to the finite element method

“…Indeed, these decompositions were used to prove this optimal convergence of a variety of hp methods in [51], [52], [25], [50], [73], [72], [20], [7]. These results about hp methods are particularly significant, since they show that, if h and p are chosen appropriately, the FEM solution is uniformly accurate as k → ∞ and the total number of degrees of freedom is proportional to k d ; i.e., the hp-FEM does not suffer from the so-called pollution effect (i.e.…”

Superconvergence analysis of linear FEM based on polynomial preserving recovery for Helmholtz equation with high wave number

“…However, if invoking the mesh condition k(kh) 2 ≤ C 0 , we have G h u h − ∇u h 0 kh(1 + C 0 )C u,g , which indicates that the pollution error between these two quantities are "almost" cancelled. This problem has been computed in [18,15,41,16] by the finite element method, the continuous interior penalty finite element method and the interior penalty discontinuous Galerkin method on both triangular meshes and rectangular meshes.…”

“…For more pre-asymptotic error estimates, Please refer to [24,25] and [8,37] for classical finite element methods as well as interior penalty finite element methods. For other methods solving the Helmholtz problems, such as the interior penalty discontinuous Galerkin method or the source transfer domain decomposition method, one can read [23,17,18,36,15,10].…”