A finite-volume implicit, unsteady, multiblock, multigrid, upwind solver, and structured multiblock grid generator for rotors are presented, and applied to lifting rotors in forward flight. These flows are particularly expensive to compute as the accurate capture of the detailed vortical wake requires fine meshes away from the blades and, hence, a parallel version of the code has been developed allowing the use of very fine meshes. Parallel performance of the code is presented, and grid dependence of the computed blade loads and wake analysed, by considering wake capturing, total blade load and sectional load variation around the azimuth. It is demonstrated that the vortical wake capture is severely influenced by grid density, and even with 32 million points grid convergence is not approached. It is also shown that if blade loads only are of interest the vorticity dissipation is not a severe problem, and coarser meshes can be used. However, if more detail is required, for example blade-vortex interaction or aero-acoustic analysis, it would appear very difficult to capture the wake to any reasonable accuracy.

The main aim of this paper is to study the error estimates of a nonconforming finite element for general second order problems, in particular, the superconvergence properties under anisotropic meshes. Some extrapolation results on rectangular meshes are also discussed. Finally, numerical results are presented, which coincides with our theoretical analysis perfectly.

It is proven that in dimension one the piecewise linear best $L^1$-approximation to the linear transport equation equipped with a set of ill-posed boundary conditions converges in $W_{loc}^{1,1}$ to the viscosity solution of the equation and the boundary layer associated with the ill-posed boundary condition is always localized in one mesh cell, i.e., the "last" one.

We examine how the wavenumber influences the compression in a wavelet boundary element method for the Helmholtz equation. We show that for wavelets with high vanishing moments the number of nonzeros in the resulting compressed matrix is approximately proportional to the square of the wavenumber. When the wavenumber is fixed, the wavelet boundary element method as optimal complexity with respect to the number of unknowns. When the mesh spacing is proportional to the wavelength, the complexity of the wavelet boundary element method is approximately proportional to the square of the number of unknowns.

In this paper we give details of a new numerical method for the solution of a singular integral equation of Volterra type that has an infinite class of solutions. The split-interval method we have adopted utilises a simple robust numerical method over an initial time interval (which includes the singularity) combined with extrapolation. We describe the method and give details of its order of convergence together with examples that show its effectiveness.

In this paper we propose a variant of a binary level set approach for solving elliptic problems with piecewise constant coefficients. The inverse problem is solved by a variational augmented Lagrangian approach with a total variation regularisation. In the binary formulation, the sought interfaces between the domains with different values of the coefficient are represented by discontinuities of the level set functions. The level set functions shall only take two discrete values, i.e. 1 and -1, but the minimisation functional is smooth. Our formulation can, under moderate amount of noise in the observations, recover rather complicated geometries without requiring any initial curves of the geometries, only a reasonable guess of the constant levels is needed. Numerical results show that our implementation of this formulation has a faster convergence than the traditional level set formulation used on the same problems.

In this paper, we generalize well-known results for the $L^2$-norm a posteriori error estimation of finite element methods applied to linear elliptic problems in convex polygonal domains to the case where the polygons are non-convex. An important factor in our analysis is the investigation of a suitable dual problem whose solution, due to the non-convexity of the domain, may exhibit corner singularities. In order to describe this singular behavior of the dual solution certain weighted Sobolev spaces are employed. Based on this framework, upper and lower a posteriori error estimates in weighted $L^2$-norms are derived. Furthermore, the performance of the proposed error estimators is illustrated with a series of numerical experiments.

We study the optimal design problem of acoustic liner to minimize fan noise radiation from commercial aircraft engine nacelles. Specifically we treat the liner impedance factor as a parameter and seek to estimate its optimal value that minimizes far-field radiated noise. The existence of such an optimal parameter is proved under the assumption that the Helmholtz equation governs the noise field. We also present numerical results to demonstrate that the choice of the optimal liner impedance factor does result in significant reduction of noise level in the far-field.

This work develops an $\varepsilon$-uniform finite element method for singularly perturbed two-point boundary value problems. A surprising and remarkable observation is illustrated: By inserting one node arbitrarily in any element, the new finite element solution always intersects with the original one at fixed points, and the errors at those points converge at the same rate as regular boundary value problems (without boundary layers). Using this fact, an effective $\varepsilon$-uniform approximation out of boundary layer is proposed by adding one point only in the element that contains the boundary layer. The thickness of the boundary layer need not be known a priori. Numerical results are carried out and compared to the Shishkin mesh for demonstration purpose.

An invalid assumption on the equivalence of alternative sets of degrees of freedom for the rotated $\mathbb{Q}_1$ element led to an incompatibility between the analysis and the application in our earlier article [1]. We outline minor modifications needed to restore the validity of all results and conclusions in the previous article.