The concept of multilevel filtering (MF) preconditioning applied to second-order selfadjoint elliptic problems is briefly reviewed. It is then shown how to effectively apply this concept to other elliptic problems such as the second-order anisotropic problem, biharmonic equation, equations on locally refined grids and interface operators arising from domain decomposition methods. Numerical resu...

Optimizing eigenvalues of biharmonic equations appears in the frequency control based on density distribution of composite rods and thin plates with clamped or simply supported boundary conditions. In this paper, we use a rearrangement algorithm to find the optimal density distribution which minimizes a specific eigenvalue. We answer the open question regarding optimal density configurations to...

Martin COSTABEL & Monique DAUGE Abstract. The 2 2 system of integral equations corresponding to the biharmonic single layer potential in R2 is known to be strongly elliptic. It is also known to be positive definite on a space of functions orthogonal to polynomials of degree one. We study the question of its unique solvability without this orthogonality condition. To each curve , we associate a ...

In this paper, we consider the well known Morley nonconforming element approximation of a nonlinear biharmonic equation which is related to the well-known two-dimensional Navier–Stokes equations. Firstly, optimal energy and H1-norm estimates are obtained. Secondly, a two-level additive Schwarz method is presented for the discrete nonlinear algebraic system. It is shown that if the Reynolds numb...

‎using variational arguments‎, ‎we prove the existence of infinitely‎ ‎many solutions to a class of $p$-biharmonic equation in‎ ‎$mathbb{r}^n$‎. ‎the existence of‎ ‎nontrivial‎ ‎solution is established under a new‎ ‎set of hypotheses on the potential $v(x)$ and the weight functions‎ ‎$h_1(x)‎, ‎h_2(x)$‎.

In this paper biharmonic maps between doubly warped product manifolds are studied. We show that the inclusion maps of Riemannian manifolds B and F into the doubly warped product f B ×b F can not be proper biharmonic maps. Also we analyze the conditions for the biharmonicity of projections f B ×b F → B and f B ×b F → F , respectively. Some characterizations for non-harmonic biharmonic maps are g...

For most `nice' elliptic boundary value problems there is a general expectation that the rst eigenfunction is unique and of xed sign. And indeed, for second order elliptic di erential equations with Dirichlet boundary conditions such a result holds as a consequence of the maximum principle. It is well known that such a maximum principle does not have a direct generalization to higher order elli...

For most `nice' elliptic boundary value problems there is a general expectation that the ̄rst eigenfunction is unique and of ̄xed sign. And indeed, for second order elliptic di®erential equations with Dirichlet boundary conditions such a result holds as a consequence of the maximum principle. It is well known that such a maximum principle does not have a direct generalization to higher order elli...