A Galerkin boundary element method for high frequency scattering by convex polygons

In this paper we consider the problem of time-harmonic acoustic scattering in two dimensions by convex polygons. Standard boundary or finite element methods for acoustic scattering problems have a computational cost that grows at least linearly as a function of the frequency of the incident wave. Here we present a novel Galerkin boundary element method, which uses an approximation space consisting of the products of plane waves with piecewise polynomials supported on a graded mesh, with smaller elements closer to the corners of the polygon. We prove that the best approximation from the approximation space requires a number of degrees of freedom to achieve a prescribed level of accuracy that grows only logarithmically as a function of the frequency. Numerical results demonstrate the same logarithmic dependence on the frequency for the Galerkin method solution. Our boundary element method is a discretization of a well-known second kind combined-layer-potential integral equation. We provide a proof that this equation and its adjoint are well-posed and equivalent to the boundary value problem in a Sobolev space setting for general Lipschitz domains.

Impact of mass lumping on gravity and Rossby waves in 2D finite-element shallow-water models

The goal of this study is to evaluate the effect of mass lumping on the dispersion properties of four finite-element velocity/surface-elevation pairs that are used to approximate the linear shallow-water equations. For each pair, the dispersion relation, obtained using the mass lumping technique, is computed and analysed for both gravity and Rossby waves. The dispersion relations are compared with those obtained for the consistent schemes (without lumping) and the continuous case. The P0-P1, RT0 and P-P1 pairs are shown to preserve good dispersive properties when the mass matrix is lumped. Test problems to simulate fast gravity and slow Rossby waves are in good agreement with the analytical results.

Trace element partitioning and soil particle characterisation around mining and smelting areas at Tharsis, Riotinto and Huelva, SW Spain

Trace elements may present an environmental hazard in the vicinity of mining and smelting activities. However, the factors controlling trace element distribution in soils around ancient and modem mining and smelting areas are not always clear. Tharsis, Riotinto and Huelva are located in the Iberian Pyrite Belt in SW Spain. Tharsis and Riotinto mines have been exploited since 2500 B.C., with intensive smelting taking place. Huelva, established in 1970 and using the Flash Furnace Outokumpu process, is currently one of the largest smelter in the world. Pyrite and chalcopyrite ore have been intensively smelted for Cu. However, unusually for smelters and mines of a similar size, the elevated trace element concentrations in soils were found to be restricted to the immediate vicinity of the mines and smelters, being found up to a maximum of 2 kin from the mines and smelters at Tharsis, Riotinto and Huelva. Trace element partitioning (over 2/3 of trace elements found in the residual immobile fraction of soils at Tharsis) and soil particles examination by SEM-EDX showed that trace elements were not adsorbed onto soil particles, but were included within the matrix of large trace element-rich Fe silicate slag particles (i.e. 1 min circle divide at least 1 wt.% As, Cu and Zn, and 2 wt.% Pb). Slag particle large size (I mm 0) was found to control the geographically restricted trace element distribution in soils at Tharsis, Riotinto and Huelva, since large heavy particles could not have been transported long distances. Distribution and partitioning indicated that impacts to the environment as a result of mining and smelting should remain minimal in the region. (c) 2006 Elsevier B.V. All rights reserved.

A preliminary investigation into mining and smelting impacts on trace element concentrations in the soils and vegetation around Tharsis, SW Spain

Toxic trace elements present an environmental hazard in the vicinity of mining and smelting activities. However. the processes of transfer of these elements to groundwater and to plants are not always clear. Tharsis mine. in the Iberian pyrite belt (SW Spain), has been exploited since 2500 BC, with extensive smelting, taking place front the 1850S until the 1920s. Sixty four soil (mainly topsoils) and vegetation samples were collected in February 2001 and analysed by ICP-AES for 23 elements. Concentrations are 6-6300 mg kg(-1) As and 14-24800 mg kg(-1) Pb in soils, and 0.20-9 mg kg(-1) As and 2-195 mg Pb in vegetation. Trace element concentrations decrease rapidly away from the mine. with As and Pb concentrations in the range 6-1850 mg kg(-1) (median 22 mg kg(-1)) and 14-31 mg, kg(-1) (median 43 mg, kg(-1)), respectively, 1 km away from the mine. These concentrations are low when compared to other well-studied mining and smelting areas (e.g. 600 mg kg(-1) As at 8 km from Yellowknife smelter, Canada; >100 mg kg(-1) Pb over 270 km(2) around the Pb-Zn Port Pirie smelter. South Australia: mean of 1419 mg kg(-1) Pb around Aberystwyth smelter, Wales, UK). The high metal content of the vegetation and the low soil pH (mean pH 4.93) indicate the potential for trace element mobility which Could explain the relatively low concentration of metals in Tharsis topsoils and cause threats to plans to redevelop the Tharsis area as an orange plantation.

Trace element geochemistry from the Birrimian metasediments of the Northern Region of Ghana

Trace element distributions in rock, soil and groundwater from of the Birrimian metasediments and granites located in the Northern Region of Ghana are described. High positive correlations are observed between selected major elements and trace metals (e.g. K2O and Rb, Al2O3 and V, Fe2O3 and V, and K2O and Y) in rocks and soils, and attributed to the presence of major source minerals. Ca and Sr were strongly correlated in groundwater, suggesting greater water-rock interaction. Low association of V with Fe is explained by (i) relatively higher mobility of V as against Fe; (ii) low Fe content in the parent rocks and (iii) variable sources of Fe and V.

A wavenumber independent boundary element method for an acoustic scattering problem

In this paper we consider the impedance boundary value problem for the Helmholtz equation in a half-plane with piecewise constant boundary data, a problem which models, for example, outdoor sound propagation over inhomogeneous. at terrain. To achieve good approximation at high frequencies with a relatively low number of degrees of freedom, we propose a novel Galerkin boundary element method, using a graded mesh with smaller elements adjacent to discontinuities in impedance and a special set of basis functions so that, on each element, the approximation space contains polynomials ( of degree.) multiplied by traces of plane waves on the boundary. We prove stability and convergence and show that the error in computing the total acoustic field is O( N-(v+1) log(1/2) N), where the number of degrees of freedom is proportional to N logN. This error estimate is independent of the wavenumber, and thus the number of degrees of freedom required to achieve a prescribed level of accuracy does not increase as the wavenumber tends to infinity.

A high-wavenumber boundary-element method for an acoustic scattering problem

In this paper we show stability and convergence for a novel Galerkin boundary element method approach to the impedance boundary value problem for the Helmholtz equation in a half-plane with piecewise constant boundary data. This problem models, for example, outdoor sound propagation over inhomogeneous flat terrain. To achieve a good approximation with a relatively low number of degrees of freedom we employ a graded mesh with smaller elements adjacent to discontinuities in impedance, and a special set of basis functions for the Galerkin method so that, on each element, the approximation space consists of polynomials (of degree $\nu$) multiplied by traces of plane waves on the boundary. In the case where the impedance is constant outside an interval $[a,b]$, which only requires the discretization of $[a,b]$, we show theoretically and experimentally that the $L_2$ error in computing the acoustic field on $[a,b]$ is ${\cal O}(\log^{\nu+3/2}|k(b-a)| M^{-(\nu+1)})$, where $M$ is the number of degrees of freedom and $k$ is the wavenumber. This indicates that the proposed method is especially commendable for large intervals or a high wavenumber. In a final section we sketch how the same methodology extends to more general scattering problems.

How to save a bad element with weak boundary conditions

The P-1-P-1 finite element pair is known to allow the existence of spurious pressure (surface elevation) modes for the shallow water equations and to be unstable for mixed formulations. We show that this behavior is strongly influenced by the strong or the weak enforcement of the impermeability boundary conditions. A numerical analysis of the Stommel model is performed for both P-1-P-1 and P-1(NC)-P-1 mixed formulations. Steady and transient test cases are considered. We observe that the P-1-P-1 element exhibits stable discrete solutions with weak boundary conditions or with fully unstructured meshes. (c) 2005 Elsevier Ltd. All rights reserved.

Finite element approximation of a sixth order nonlinear degenerate parabolic equation

We consider a finite element approximation of the sixth order nonlinear degenerate parabolic equation ut = ?.( b(u)? 2u), where generically b(u) := |u|? for any given ? ? (0,?). In addition to showing well-posedness of our approximation, we prove convergence in space dimensions d ? 3. Furthermore an iterative scheme for solving the resulting nonlinear discrete system is analysed. Finally some numerical experiments in one and two space dimensions are presented.