A discretized boundary value problem for the Laplace equation with the Dirichlet and Neumann boundary conditions on an equilateral triangle with a triangular mesh is transformed into a problem of the same type on a rectangle. Explicit formulae for all eigenvalues and all eigenfunctions are given.
The electrowetting process is commonly used to handle very small amounts of liquid on a solid surface. This process can be modelled mathematically with the help of the shape optimization theory. However, solving numerically the resulting shape optimization problem is a very complex issue, even for reduced models that occur in simplified geometries. Recently, the second author obtained convincing results in the 2D axisymmetric case. In this paper, we propose and analyze a method that is suitable...
The accuracy of the domain embedding method from [A. Rieder, Modél. Math. Anal. Numér.32 (1998) 405-431] for the solution of Dirichlet problems suffers under a coarse boundary approximation. To overcome this drawback the method is furnished with an a priori (static) strategy for an adaptive approximation space refinement near the boundary. This is done by selecting suitable wavelet subspaces. Error estimates and numerical experiments validate the proposed adaptive scheme. In contrast to similar,...
We present a numerical algorithm to solve the micromagnetic equations based on tangential-plane minimization for the magnetization update and a homothethic-layer decomposition of outer space for the computation of the demagnetization field. As a first application, detailed results on the flower-vortex transition in the cube of Micromagnetic Standard Problem number 3 are obtained, which confirm, with a different method, those already present in the literature, and validate our method and code. We...
We present a numerical algorithm to solve the micromagnetic equations based on tangential-plane minimization for the magnetization update and a homothethic-layer decomposition of outer space for the computation of the demagnetization field. As a first application, detailed results on the flower-vortex transition in the cube of Micromagnetic Standard Problem number 3 are obtained, which confirm, with a different method, those already present in the literature, and validate our method and...
Les méthodes sans maillage emploient une interpolation associée à un ensemble de particules : aucune information concernant la connectivité ne doit être fournie. Un des atouts de ces méthodes est que la discrétisation peut être enrichie d’une façon très simple, soit en augmentant le nombre de particules (analogue à la stratégie de raffinement ), soit en augmentant l’ordre de consistance (analogue à la stratégie de raffinement ). Néanmoins, le coût du calcul des fonctions d’interpolation est très...
Les méthodes sans maillage emploient une interpolation associée à un ensemble de particules : aucune information concernant la connectivité ne doit être fournie. Un des atouts de ces méthodes est que la discrétisation peut être enrichie d'une façon très simple, soit en augmentant le nombre de particules (analogue à la stratégie de raffinement h), soit en augmentant l'ordre de consistance (analogue à la stratégie de raffinement p). Néanmoins, le coût du calcul des fonctions d'interpolation est...
We consider the lowest-order Raviart–Thomas mixed finite element method for second-order elliptic problems on simplicial meshes in two and three space dimensions. This method produces saddle-point problems for scalar and flux unknowns. We show how to easily and locally eliminate the flux unknowns, which implies the equivalence between this method and a particular multi-point finite volume scheme, without any approximate numerical integration. The matrix of the final linear system is sparse, positive...