Consistent dirichlet boundary conditions for numerical solution of moving boundary problems

We consider the imposition of Dirichlet boundary conditions in the finite element mod-elling of moving boundary problems in one and two dimensions for which the total mass is prescribed. A modification of the standard linear finite element test space allows the boundary conditions to be imposed strongly whilst simultaneously conserving a discrete mass. The validity of the technique is assessed for a specific moving mesh finite element method, although the approach is more general. Numerical comparisons are carried out for mass-conserving solutions of the porous medium equation with Dirichlet boundary conditions and for a moving boundary problem with a source term and time-varying mass

Equation of motion approach to the Hubbard model in infinite dimensions

We consider the Hubbard model on the infinite-dimensional Bethe lattice and construct a systematic series of self-consistent approximations to the one-particle Green's function, $G^{(n)}(\omega),\ n=2,3,\dots\$ . The first $n-1$ equations of motion are exactly fullfilled by $G^{(n)}(\omega)$ and the $n$'th equation of motion is decoupled following a simple set of decoupling rules. $G^{(2)}(\omega)$ corresponds to the Hubbard-III approximation. We present analytic and numerical results for the Mott-Hubbard transition at half filling for $n=2,3,4$.Comment: 10pager, REVTEX, 8-figures not available in postscript, manuscript may be understood without figure

Glacial isostatic adjustment associated with the Barents Sea ice sheet: a modelling inter-comparison

The 3D geometrical evolution of the Barents Sea Ice Sheet (BSIS), particularly during its late-glacial retreat phase, remains largely ambiguous due to the paucity of direct marine- and terrestrial-based evidence constraining its horizontal and vertical extent and chronology. One way of validating the numerous BSIS reconstructions previously proposed is to collate and apply them under a wide range of Earth models and to compare prognostic (isostatic) output through time with known relative sea-level (RSL) data. Here we compare six contrasting BSIS load scenarios via a spherical Earth system model and derive a best-fit, χ2 parameter using RSL data from the four main terrestrial regions within the domain: Svalbard, Franz Josef Land, Novaya Zemlya and northern Norway. Poor χ2 values allow two load scenarios to be dismissed, leaving four that agree well with RSL observations. The remaining four scenarios optimally fit the RSL data when combined with Earth models that have an upper mantle viscosity of 0.2–2 × 1021 Pa s, while there is less sensitivity to the lithosphere thickness (ranging from 71 to 120 km) and lower mantle viscosity (spanning 1–50 × 1021 Pa s). GPS observations are also compared with predictions of present-day uplift across the Barents Sea. Key locations where relative sea-level and GPS data would prove critical in constraining future ice-sheet modelling efforts are also identified

Scale-invariant moving finite elements for nonlinear partial differential equations in two dimensions

A scale-invariant moving finite element method is proposed for the adaptive solution of nonlinear partial differential equations. The mesh movement is based on a finite element discretisation of a scale-invariant conservation principle incorporating a monitor function, while the time discretisation of the resulting system of ordinary differential equations is carried out using a scale-invariant time-stepping which yields uniform local accuracy in time. The accuracy and reliability of the algorithm are successfully tested against exact self-similar solutions where available, and otherwise against a state-of-the-art h-refinement scheme for solutions of a two-dimensional porous medium equation problem with a moving boundary. The monitor functions used are the dependent variable and a monitor related to the surface area of the solution manifold

Towards analytical approaches to the dynamical-cluster approximation

I introduce several simplified schemes for the approximation of the self-consistency condition of the dynamical cluster approximation. The applicability of the schemes is tested numerically using the fluctuation-exchange approximation as a cluster solver for the Hubbard model. Thermodynamic properties are found to be practically indistinguishable from those computed using the full self-consistent scheme in all cases where the non-interacting partial density of states is replaced by simplified analytic forms with matching 1st and 2nd moments. Green functions are also compared and found to be in close agreement, and the density of states computed using Pad\'{e} approximant analytic continuation shows that dynamical properties can also be approximated effectively. Extensions to two-particle properties and multiple bands are discussed. Simplified approaches to the dynamical cluster approximation should lead to new analytic solutions of the Hubbard and other models

Semiclassical Analysis of Extended Dynamical Mean Field Equations

The extended Dynamical Mean Field Equations (EDMFT) are analyzed using semiclassical methods for a model describing an interacting fermi-bose system. We compare the semiclassical approach with the exact QMC (Quantum Montecarlo) method. We found the transition to an ordered state to be of the first order for any dimension below four.Comment: RevTex, 39 pages, 16 figures; Appendix C added, typos correcte