Vekua theory for the Helmholtz operator

Abstract

Abstract. Vekua operators map harmonic functions defined on domain in R 2 to solutions of elliptic partial differential equations on the same domain and vice versa. In this paper, following the original work of I. Vekua (Ilja Vekua (1907-1977, Soviet-Georgian mathematician), we define Vekua operators in the case of the Helmholtz equation in a completely explicit fashion, in any space dimension N ≥ 2. We prove (i) that they actually transform harmonic functions and Helmholtz solutions into each other; (ii) that they are inverse to each other; and (iii) that they are continuous in any Sobolev norm in star-shaped Lipschitz domains. Finally, we define and compute the generalized harmonic polynomials as the Vekua transforms of harmonic polynomials. These results are instrumental in proving approximation estimates for solutions of the Helmholtz equation in spaces of circular, spherical, and plane waves.

Approximation by harmonic polynomials in star-shaped domains and exponential convergence of Trefftzhp-dGFEM

“…We chose to consider star-shaped domains since, as mentioned above, the main application we have in mind involves the use of the Vekua operators, which are defined only under this assumption, see [31]. On the other hand, all the proofs in the present article would be hugely simplified if we assumed convex domains instead (e.g.…”

“…The monotonicity in dependence of ε and R 2 can be verified by computing the derivative of the expression in (31). The last inequality in the assertion follows from η(ε, R 1 , R 2 ) < lim ε→∞ η(ε, R 1 , R 2 ) = 2 π arccos 1 − R 2 1 R 2 2 = 2 π arcsin R 1 R 2 which uses the monotonicity of η as a function of ε, and the identity sin arccos √ 1 − t 2 = |t|.…”

“…Fix ε > 0. Define an integer N ∈ N such that η ε := 2 N ≤ η(ε, ρ 0 , 1 − ρ), where η(·, ·, ·) was defined in formula (31). We select the points w j ∈ ∂D, j = 1, .…”

“…The results of the present paper can be extended to general second-order elliptic equations by means of the socalled Vekua theory [31,36]. This technique provides continuous bijections between spaces of harmonic functions and spaces of solutions to the considered elliptic equation.…”

Plane wave approximation of homogeneous Helmholtz solutions

“…In particular, they take harmonic polynomials to generalized harmonic polynomials. Using the continuity of Vekua operators [18], approximation estimates for homogeneous Helmholtz solutions in the spaces HP ω,L (R N ) can be obtained from approximation estimates of harmonic functions by harmonic polynomials. In [12], we have proved h-version approximation estimates for harmonic functions by harmonic polynomials in any space dimension, using a simple Bramble-Hilbert argument.…”