Analysis of a coupled finite-infinite element method for exterior Helmholtz problems.

*(English)*Zbl 1013.65122The authors analyse convergence of the infinite element method (IEM) for exterior Helmholtz problems when a coupled finite-infinite element method is applied. If \(\Omega ^+\) is the exterior of a bounded domain \(\Omega \) in three dimensional Euclidean space, with a sufficiently smooth boundary \(\Gamma \), a smooth artificial boundary \(\Gamma _a \subset \Omega ^+\) enclosing \(\Omega \) is introduced. The resulting annular domain (bounded by \(\Gamma \) and \(\Gamma _a\)) and its exterior unbounded component are, respectively, denoted by \(\Omega _a\) and \(\Omega _a^+\). A triangulation of \(\Omega _a\) and a corresponding finite element (FE) space of conforming \(h-p\) elements induce triangulations and spaces in \(\Gamma _a\) and \(\Omega _a^+\). This coupling allows to look for a joint FE-IE approximation to the solution of the original problem.

The paper is concerned with the influence of the IEM approximation on the convergence of the coupled approximate solution. To achieve this, the introduction of a Dirichlet-to-Neumann (DtN) operator from \(H^{1/2}(\Gamma _a)\) into \(H^{-1/2}(\Gamma _a)\) allows to transform the original (weak) analytic problem on \(\Omega ^+\) into a weak problem on \(\Omega _a\). Also, approximate DtN operators transform the IE formulation on \(\Omega _a^+\) into a finite dimensional variational formulation in \(H^1(\Omega _a)\). By considering convenient enlargements of the (finite dimensional) approximating spaces, the IEM error is decomposed as \[ u-u_h^N = (u-u^N) + (u^N-u_h^N) , \] where \(u_h^N\) is the solution of the variational (weak) formulation in the finite element space on \(\Omega _a\cup \Gamma _a\) and \(u^N\) is the solution of the variational formulation in an \(N-\)dimensional subspace of the enlargements. The paper focuses on the term \(u-u^N\) and by assuming an \(inf-sup\) condition of the weak formulation a bound is given for \(\|u-u^N\|_1\), involving the operator norm of the difference between the DtN operator and its approximation in the already mentioned subspace.

The paper then concentrates on the case where \(\Gamma _a\) is a sphere. A uniform \(inf-sup\) condition is shown to hold. Also well-posedness, i.e. existence and uniqueness, is proved for the continuous and discrete formulations involving the DtN operators, which easily yields a proof of existence-uniqueness for the original exterior problem. It is also proved that algebraic convergence rates of any order can be achieved, which is shown to imply exponential convergence, that is, there exist positive constants \(C\) and \(\kappa \) such that \[ \|u-u_N \|_1 \leq C e^{-\kappa N} . \]

The paper is concerned with the influence of the IEM approximation on the convergence of the coupled approximate solution. To achieve this, the introduction of a Dirichlet-to-Neumann (DtN) operator from \(H^{1/2}(\Gamma _a)\) into \(H^{-1/2}(\Gamma _a)\) allows to transform the original (weak) analytic problem on \(\Omega ^+\) into a weak problem on \(\Omega _a\). Also, approximate DtN operators transform the IE formulation on \(\Omega _a^+\) into a finite dimensional variational formulation in \(H^1(\Omega _a)\). By considering convenient enlargements of the (finite dimensional) approximating spaces, the IEM error is decomposed as \[ u-u_h^N = (u-u^N) + (u^N-u_h^N) , \] where \(u_h^N\) is the solution of the variational (weak) formulation in the finite element space on \(\Omega _a\cup \Gamma _a\) and \(u^N\) is the solution of the variational formulation in an \(N-\)dimensional subspace of the enlargements. The paper focuses on the term \(u-u^N\) and by assuming an \(inf-sup\) condition of the weak formulation a bound is given for \(\|u-u^N\|_1\), involving the operator norm of the difference between the DtN operator and its approximation in the already mentioned subspace.

The paper then concentrates on the case where \(\Gamma _a\) is a sphere. A uniform \(inf-sup\) condition is shown to hold. Also well-posedness, i.e. existence and uniqueness, is proved for the continuous and discrete formulations involving the DtN operators, which easily yields a proof of existence-uniqueness for the original exterior problem. It is also proved that algebraic convergence rates of any order can be achieved, which is shown to imply exponential convergence, that is, there exist positive constants \(C\) and \(\kappa \) such that \[ \|u-u_N \|_1 \leq C e^{-\kappa N} . \]

Reviewer: Juan Pedro Milaszewicz (Buenos Aires)

##### MSC:

65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |

65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |

35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |