During the development of a parallel solver for Maxwell equations by integral formulations and Fast Multipole Method (FMM), we needed to optimize a critical part including a lot of communications and computations. Generally, many parallel programs need to communicate, but choosing explicitly the way and the instant may decrease the efficiency of the overall program. So, the overlapping of computations and communications may be a way to reduce this drawback. We will see a implementation of this techniques...

During the development of a parallel solver for Maxwell equations by integral formulations and Fast Multipole Method (FMM), we needed to optimize a critical part including a lot of communications and computations. Generally, many parallel programs need to communicate, but choosing explicitly the way and the instant may decrease the efficiency of the overall program. So, the overlapping of computations and communications may be a way to reduce this drawback. We will see a implementation of this...

Let $G\subset {ℝ}^m$$(m\ge 2)$ be an open set with a compact boundary $B$ and let $\sigma \ge 0$ be a finite measure on $B$. Consider the space $L^1\left(\sigma \right)$ of all $\sigma$-integrable functions on $B$ and, for each...

Let $K\subset {ℝ}^m$ ($m\ge 2$) be a compact set; assume that each ball centered on the boundary $B$ of $K$ meets $K$ in a set of positive Lebesgue measure. Let $C_0^{\left(1\right)}$ be the class of all continuously differentiable real-valued functions with compact support in ${ℝ}^m$ and denote by ${\sigma }_m$ the area of the unit sphere in ${ℝ}^m$. With each $\varphi \in C_0^{\left(1\right)}$ we associate the function $$W_K\varphi \left(z\right)=\frac{1}{{\sigma }_m}\underset{{ℝ}^m\setminus K}{\to }\int \mathrm{g}rad\varphi \left(x\right)·\frac{z-x}{{|z-x|}^m}x$$ of the variable $z\in K$ (which is continuous in $K$ and harmonic in $K\setminus B$). $W_K\varphi$ depends only on the restriction ${\varphi |}_B$ of $\varphi$ to the boundary $B$ of $K$. This gives rise to a linear operator $W_K$ acting from...

A new and elegant procedure is proposed for the solution of mixed potential problems in a half-space with a circular line of division of boundary conditions. The approach is based on a new type of integral operators with special properties. Two general external problems are solved; i) An arbitrary potential is specified at the boundary outside a circle, and its normal derivative is zero inside; ii) An arbitrary normal derivative is given outside the circle, and be potential is zero inside. Several...

In this paper some embedding theorems related to fractional integration and differentiation in harmonic mixed norm spaces $h(p,q,\alpha )$ on the half-space are established. We prove that mixed norm is equivalent to a “fractional derivative norm” and that harmonic conjugation is bounded in $h(p,q,\alpha )$ for the range $0<p\le \infty$, $0<q\le \infty$. As an application of the above, we give a characterization of $h(p,q,\alpha )$ by means of an integral representation with the use of Besov spaces.

In this paper we prove that a subharmonic function in ℝm of finite λ-type can be represented (within some subharmonic function) as the sum of a generalized Weierstrass canonical integral and a function of finite λ-type which tends to zero uniformly on compacts of ℝm. The known Brelot-Hadamard representation of subharmonic functions in ℝm of finite order can be obtained as a corollary from this result. Moreover, some properties of R-remainders of λ-admissible mass distributions are investigated.

The $H$-cone is an abstract model for the cone of positive superharmonic functions on a harmonic space or for the cone of excessive functions with respect to a resolvent family, having sufficiently many properties in order to develop a good deal of balayage theory and also to construct a dual concept which is also an $H$-cone. There are given an integral representation theorem and a representation theorem as an $H$-cone of functions for which fine topology, thinnes, negligible sets and the sheaf property...

We derive various integral representation formulas for a function minus a polynomial in terms of vector field gradients of the function of appropriately high order. Our results hold in the general setting of metric spaces, including those associated with Carnot-Carathéodory vector fields, under the assumption that a suitable $L^1$ to $L^1$ Poincaré inequality holds. Of particular interest are the representation formulas in Euclidean space and stratified groups, where polynomials exist and $L^1$ to $L^1$ Poincaré...

We generalize the notions of harmonic conjugate functions and Hilbert transforms to higher-dimensional euclidean spaces, in the setting of differential forms and the Hodge-Dirac system. These harmonic conjugates are in general far from being unique, but under suitable boundary conditions we prove existence and uniqueness of conjugates. The proof also yields invertibility results for a new class of generalized double layer potential operators on Lipschitz surfaces and boundedness of related Hilbert...

Let ${\Omega }_1,...,{\Omega }_n$ be harmonic spaces of Brelot with countable base of completely determining domains. The elements of a subcone $\mathbf{C}$ of the cone of positive $n$-superharmonic functions in ${\Omega }_1\times ...\times {\Omega }_n$ is shown to have an integral representation with the aid of Radon measures on the extreme elements belonging to a compact base of $\mathbf{C}$. The extreme elements are shown to be the product of extreme superharmonic functions on the component spaces and the measure representing each element is shown to be unique. Necessary and sufficient...