Differential operator

Summary

In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and returns another function (in the style of a higher-order function in computer science). This article considers mainly linear differential operators, which are the most common type. However, non-linear differential operators also exist, such as the Schwarzian derivative. Definition Given a nonnegative integer m, an order-m linear differential operator is a map P from a function space \mathcal{F}_1 to another function space \mathcal{F}_2 that can be written as: P = \sum_{|\alpha|\le m}a_\alpha(x) D^\alpha\ , where \alpha = (\alpha_1,\alpha_2,\cdots,\alpha_n) is a multi-index of non-negative integers, |\alpha| = \alpha_1 + \alpha_2 + \cdots +

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Inversion de Fourier ponctuelle

Emanuel Enrico Seifert

Let f be an integrable function on RN, a a point in RN and B a complex number. If the mean value of f on the sphere of centre a and radius r tends to B when r tends to 0, we show that the Fourier integral at a of f is summable to B in Cesàro means of order λ > (N-1)/2. Let now U be a bounded open subset of RN whose boundary ∂U is a real analytic submanifold of RN with dimension N-1. We deduce from the preceding result that the Fourier integral at a of the indicator function of U is summable in Cesàro means of order λ > (N-1)/2 to 1 if a ∈ U, to 1/2 if a ∈ ∂U and to 0 if a ∉ U. We then show that if the function defined on ∂U by y → ‖ y - a ‖ has only a finite number of critical points, then we can take λ less or equal to (N-1)/2 ; more precisely, it suffices to have λ > (N-3)/2 + σ(a|∂U), where σ (a|∂U) < 0 is the maximum of the oscillatory indices associated to the critical points of y → ‖ y - a ‖ ; this generalizes results obtained by Pinsky, Taylor and Popov in 1997. Finally, writing μ∂U for the natural measure supported by ∂U, P(D) for a differential operator with constant coefficients of order m and b for a C∞ function on RN, we show that, if a is a point outside ∂U such that ‖ y - a ‖ has only a finite number of critical points on ∂U, the Fourier integral at a of the distribution P(D) bμ∂U is summable to 0 in Cesàro means of order λ > (N-1)/2 + m + σ (a|∂U) ; this generalizes a result obtained by Gonzàlez Vieli in 2002.

EPFL2007

Multi-level preconditioner for solving the Navier-Stokes equations in hemodynamics applications

Dana Christen

We present a multi-level algorithm to approximate the inverse of the fluid block in a Navier-Stokes saddle-point matrix where the coarse level is defined as a restriction of the degrees of freedom to those of lower order finite elements. A one-level scheme involving P1 and P2 finite elements is studied in details and several transfer operators are compared by means of two reference problems. Numerical results show that restriction and prolongation operators based on L2 projection lead to faster GMRES convergence of the fluid part, for all the examined combinations of mesh size, time step and Reynolds number.

2013

In this thesis, we study two distinct problems. The first problem consists of studying the linear system of partial differential equations which consists of taking a k-form, and applying the exterior derivative 'd' to it and add the wedge product with a 1-form 'a'. The study of this differential operator is linked to the study of the multiplication by a two form, that is the system of linear equations where we take a k-form and apply the exterior wedge product by 'da', the exterior derivative of 'a'. We establish links between the partial differential equation and the linear system. The second problem is a generalization of the symmetric gradient and the curl equation. The equation of a symmetric gradient consists of taking a vector field, apply the gradient and then add the transpose of the gradient, whereas in the curl equation we subtract the transpose of the gradient. Both can be seen as an equation of the form A * grad u + (grad u)t * A, where A is a symmetric matrix for the case of the symmetric gradient and skew symmetric for the curl equation. We generalize to the case where A verifies no symmetry assumption and more significantly add a Dirichlet condition on the boundary.

EPFL2018