A preconditioner defined by an algebraic multigrid cycle for a damped Helmholtz operator is proposed for the Helmholtz equation. This approach is well-suited for acoustic scattering problems in complicated computational domains and with varying material properties. The spectral properties of the preconditioned systems and the convergence of the GMRES method are studied with linear, quadratic, and cubic finite element discretizations. Numerical experiments are performed with two-dimensional problems describing acoustic scattering in a cross section of a car cabin and in a layered medium. Asymptotically the number of iterations grows linearly with respect to the frequency while for lower frequencies the growth is milder. The proposed preconditioner is particularly effective for low-frequency and mid-frequency problems.

Fixed domain methods have well-known advantages in the solution of variable domain problems including inverse interface problems. This paper examines two new control approaches to optimal design problems governed by general elliptic boundary value problems with Dirichlet boundary conditions. Numerical experiments are also included.

We consider a controllability technique for the numerical solution of the Helmholtz equation. The original time-harmonic equation is represented as an exact controllability problem for the time-dependent wave equation. This problem is then formulated as a least-squares optimization problem, which is solved by the conjugate gradient method. Such an approach was first suggested and developed in the 1990s by French researchers and we introduce some improvements to its practical realization.We use higher-order spectral elements for spatial discretization, which leads to high accuracy and lumped mass matrices. Higher-order approximation reduces the pollution effect associated with finite element approximation of time-harmonic wave equations, and mass lumping makes explicit time-stepping schemes for the wave equation very efficent. We also derive a new way to compute the gradient of the least-squares functional and use algebraic multigrid method for preconditioning the conjugate gradient algorithm.Numerical results demonstrate the significant improvements in efficiency due to the higher-order spectral elements. For a given accuracy, spectral element method requires fewer computational operations than conventional finite element method. In addition, by using higher-order polynomial basis the influence of the pollution effect is reduced.

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We formulate the Helmholtz equation as an exact controllability problem for the time-dependent wave equation. The problem is then discretized in time domain with central finite difference scheme and in space domain with spectral elements. This approach leads to high accuracy in spatial discretization. Moreover, the spectral element method results in diagonal mass matrices, which makes the time integration of the wave equation highly efficient. After discretization, the exact controllability problem is reformulated as a least squares problem, which is solved by the conjugate gradient method. We illustrate the method with some numerical experiments, which demonstrate the significant improvements in efficiency due to the higher order spectral elements. For a given accuracy, the controllability technique with spectral element method requires fewer computational operations than with conventional finite element method. In addition, by using higher order polynomial basis the influence of the pollution effect is reduced.