We consider a biharmonic problem with Navier type boundary conditions , on a family of truncated sectors in of radius , and opening angle , when is close to . The family of right-hand sides is assumed to depend smoothly on in . The main result is that converges to when with respect to the -norm. We can also show that the -topology is optimal for such a convergence result.
We propose and analyze a domain decomposition method on non-matching grids for partial differential equations with non-negative characteristic form. No weak or strong continuity of the finite element functions, their normal derivatives, or linear combinations of the two is imposed across the boundaries of the subdomains. Instead, we employ suitable bilinear forms defined on the common interfaces, typical of discontinuous Galerkin approximations. We prove an error bound which is optimal with respect...
A brief survey is given to show that harmonic averages enter in a natural way in the numerical solution of various variable coefficient problems, such as in elliptic and transport equations, also of singular perturbation types. Local Green’s functions used as test functions in the Petrov-Galerkin finite element method combined with harmonic averages can be very efficient and are related to exact difference schemes.
We study the approximation properties of some finite element subspaces of H(div;Ω) and H(curl;Ω) defined on hexahedral meshes in three dimensions. This work extends results previously obtained for quadrilateral H(div;Ω) finite elements and for quadrilateral scalar finite element spaces. The finite element spaces we consider are constructed starting from a given finite dimensional space of vector fields on the reference cube, which is then transformed to a space of vector fields on a hexahedron using...
We study the approximation properties of some finite element subspaces of H(div;Ω) and H(curl;Ω) defined on hexahedral meshes in three dimensions. This work extends results previously obtained for quadrilateral H(div;Ω) finite elements and for quadrilateral scalar finite element spaces. The finite element spaces we consider are constructed starting from a given finite dimensional space of vector fields on the reference cube, which is then transformed to a space of vector fields on a hexahedron...
We propose transmission conditions of order 1, 2 and 3 approximating the shielding behaviour of thin conducting curved sheets for the magneto-quasistatic eddy current model in 2D. This model reduction applies to sheets whose thicknesses ε are at the order of the skin depth or essentially smaller. The sheet has itself not to be resolved, only its midline is represented by an interface. The computation is directly in one step with almost no additional cost. We prove the well-posedness w.r.t. to...
We propose transmission conditions of order 1, 2 and 3 approximating the shielding behaviour of thin conducting curved sheets for the magneto-quasistatic eddy current model in 2D. This model reduction applies to sheets whose thicknesses ε are at the order of the skin depth or essentially smaller. The sheet has itself not to be resolved, only its midline is represented by an interface. The computation is directly in one step with almost no additional cost. We prove the well-posedness w.r.t. to...
We consider the Brownian path-valued process studied in [LG1], [LG2], which is closely related to super Brownian motion. We obtain several potential-theoretic results related to this process. In particular, we give an explicit description of the capacitary distribution of certain subsets of the path space, such as the set of paths that hit a given closed set. These capacitary distributions are characterized as the laws of solutions of certain stochastic differential equations. They solve variational...
We are concerned with the asymptotic analysis of optimal control problems for -D partial differential equations defined on a periodic planar graph, as the period of the graph tends to zero. We focus on optimal control problems for elliptic equations with distributed and boundary controls. Using approaches of the theory of homogenization we show that the original problem on the periodic graph tends to a standard linear quadratic optimal control problem for a two-dimensional homogenized system, and...
We are concerned with the asymptotic analysis of optimal control problems for 1-D partial differential equations defined on a periodic planar graph, as the period of the graph tends to zero. We focus on optimal control problems for elliptic equations with distributed and boundary controls. Using approaches of the theory of homogenization we show that the original problem on the periodic graph tends to a standard linear quadratic optimal control problem for a two-dimensional homogenized system,...
We homogenize a class of nonlinear differential equations set in highly heterogeneous media. Contrary to the usual approach, the coefficients in the equation characterizing the material properties are supposed to be uncertain functions from a given set of admissible data. The problem with uncertainties is treated by means of the worst scenario method, when we look for a solution which is critical in some sense.