The Uncoupling of Boundary Integral and Finite Element Methods for Nonlinear Boundary Value Problems

AbstractIn this paper the uncoupling of boundary integral and finite element methods is applied to study the weak solvability of certain nonlinear exterior boundary value problems. As a model we consider a nonlinear second order elliptic equation in divergence form in a bounded inner region of the plane, which becomes the Laplace equation in the corresponding unbounded exterior region. We provide sufficient conditions for the nonlinear coefficients from which existence, uniqueness, and approximation results are established. In particular, nonlinear equations yielding both monotone and nonmonotone operators are analyzed

Helium mixtures in nanotube bundles

An analogue to Raoult's law is determined for the case of a 3He-4He mixture adsorbed in the interstitial channels of a bundle of carbon nanotubes. Unlike the case of He mixtures in other environments, the ratio of the partial pressures of the coexisting vapor is found to be a simple function of the ratio of concentrations within the nanotube bundle.Comment: 3 pages, no figures, submitted to Phys. Rev. Let

A mixed virtual element method for the Brinkman problem

In this paper, we introduce and analyze a mixed virtual element method (mixed-VEM) for the two-dimensional Brinkman model of porous media flow with non-homogeneous Dirichlet boundary conditions. More precisely, we employ a dual-mixed formulation in which the only unknown is given by the pseudostress, whereas the velocity and pressure are computed via postprocessing formulae. We first recall the corresponding variational formulation, and then summarize the main mixed-VEM ingredients that are required for our discrete analysis. In particular, in order to define a calculable discrete bilinear form, whose continuous version involves deviatoric tensors, we propose two well-known alternatives for the local projector onto a suitable polynomial subspace, which allows the explicit integration of these terms. Next, we show that the global discrete bilinear form satisfies the hypotheses required by the Lax–Milgram lemma. In this way, we conclude the well-posedness of our mixed-VEM scheme and derive the associated a priori error estimates for the virtual solution as well as for the fully computable projection of it. Furthermore, we also introduce a second element-by-element postprocessing formula for the pseudostress, which yields an optimally convergent approximation of this unknown with respect to the broken ℍ(div)-norm. Finally, several numerical results illustrating the good performance of the method and confirming the theoretical rates of convergence are presented.En este artículo, presentamos y analizamos un método de elemento virtual mixto (VEM mixto) para el modelo bidimensional Brinkman de flujo de medios porosos con condiciones de contorno de Dirichlet no homogéneas. Más precisamente, empleamos una formulación de doble mezcla en la que el pseudoesfuerzo da lo único desconocido, mientras que la velocidad y la presión se calculan mediante fórmulas de postprocesamiento. Primero recordamos la formulación variacional correspondiente y luego resumimos los principales ingredientes mixtos de VEM que se requieren para nuestro análisis discreto. En particular, para definir una forma bilineal discreta calculable, cuya versión continua involucra tensores desviadores, proponemos dos alternativas bien conocidas para el proyector local en un subespacio polinomial adecuado, que permite la integración explícita de estos términos. A continuación, mostramos que la forma bilineal discreta global satisface las hipótesis requeridas por el lema de Lax-Milgram. De esta manera, concluimos la buena posición de nuestro esquema de VEM mixto y derivamos las estimaciones de error a priori asociadas para la solución virtual, así como para la proyección totalmente computable de la misma. Además, también presentamos una segunda fórmula de posprocesamiento elemento por elemento para el pseudoesfuerzo, que produce una aproximación óptimamente convergente de esta incógnita con respecto a la forma broken (div) rota. Finalmente, se presentan varios resultados numéricos que ilustran el buen desempeño del método y confirman las tasas teóricas de convergencia.Universidad Nacional, Costa RicaEscuela de Matemátic

An augmented mixed-primal finite element method for a coupled flow-transport problem.

In this paper we analyze the coupling of a scalar nonlinear convection-diffusion problem with the Stokes equations where the viscosity depends on the distribution of the solution to the transport problem. An augmented variational approach for the fluid flow coupled with a primal formulation for the transport model is proposed. The resulting Galerkin scheme yields an augmented mixed-primal finite element method employing Raviart−Thomas spaces of order k for the Cauchy stress, and continuous piecewise polynomials of degree ≤ k + 1 for the velocity and also for the scalar field. The classical Schauder and Brouwer fixed point theorems are utilized to establish existence of solution of the continuous and discrete formulations, respectively. In turn, suitable estimates arising from the connection between a regularity assumption and the Sobolev embedding and Rellich−Kondrachov compactness theorems, are also employed in the continuous analysis. Then, sufficiently small data allow us to prove uniqueness and to derive optimal a priori error estimates. Finally, we report a few numerical tests confirming the predicted rates of convergence, and illustrating the performance of a linearized method based on Newton−Raphson iterations; and we apply the proposed framework in the simulation of thermal convection and sedimentation-consolidation processes

Some recent developments in applied functional analysis

From its early stages, the intensive development of functional analysis and the remarkable advances of its methods cannot be explained without its link with other areas of mathematics and, above all, its role as an essential framework for numerical analysis and computer simulation, PDEs, modeling realworld phenomena, variational inequalities, or optimization, just to name a few. In this special issue we highlight some aspects of functional analysis which are used in connection with other branches of mathematics or science, either as a direct application or as a theoretical result which is essential for such an application. Although it is not possible to collect here the huge production of the research activity on this vast field of modern mathematics, the selected works gather together a range of topics which reflect some of the current research on applied functional analysis: bases in Banach spaces, wavelet transforms, fixed point theory, and applications to ODEs, electronic circuit simulation, or numerical solution of PDEs, integral equations, or problems on option pricing in mathematical finance. In this way, we have achieved one of our purposes, which is the exchange of ideas among researchers working both in abstract and applied functional analysis