We recall here some theoretical results of Helffer et al. [Ann. Inst. H. Poincaré Anal. Non Linéaire (2007) doi:10.1016/j.anihpc.2007.07.004] about minimal partitions and propose numerical computations to check some of their published or unpublished conjectures and exhibit new ones.

We consider the numerical solution of diffusion problems in $(0,T)\times \Omega$ for $\Omega \subset {ℝ}^d$ and for $T\>0$ in dimension $d\ge 1$. We use a wavelet based sparse grid space discretization with mesh-width $h$ and order $p\ge 1$, and $hp$ discontinuous Galerkin time-discretization of order $r=O\left(\left|logh\right|\right)$ on a geometric sequence of $O\left(\left|logh\right|\right)$ many time steps. The linear systems in each time step are solved iteratively by $O\left(\left|logh\right|\right)$ GMRES iterations with a wavelet preconditioner. We prove that this algorithm gives an $L^2\left(\Omega \right)$-error of $O\left(N^{-p}\right)$ for $u(x,T)$ where $N$ is the total number of operations,...

We consider the numerical solution of diffusion problems in (0,T) x Ω for $\Omega \subset {ℝ}^d$ and for T > 0 in dimension dd ≥ 1. We use a wavelet based sparse grid space discretization with mesh-width h and order pd ≥ 1, and hp discontinuous Galerkin time-discretization of order $r=O\left(\left|logh\right|\right)$ on a geometric sequence of $O\left(\left|logh\right|\right)$ many time steps. The linear systems in each time step are solved iteratively by $O\left(\left|logh\right|\right)$ GMRES iterations with a wavelet preconditioner. We prove that this algorithm gives an L2(Ω)-error of O(N-p) for u(x,T)...

In this paper, we study the global convergence for the numerical solutions of nonlinear Volterra integral equations of the second kind by means of Galerkin finite element methods. Global superconvergence properties are discussed by iterated finite element methods and interpolated finite element methods. Local superconvergence and iterative correction schemes are also considered by iterated finite element methods. We improve the corresponding results obtained by collocation methods in the recent...

The paper studies mesh dependent numerical solution of groundwater problems with singularities, caused by boreholes represented as points, instead of a real radius. We show on examples, that the numerical solution of the borehole pumping problem with point source (singularity) can be related to the exact solution of a regular problem with adapted geometry of a finite borehole radius. The radius providing the fit is roughly proportional to the mesh step. Next we define a problem of fracture-rock...

Natural superconvergence of the least-squares finite element method is surveyed for the one- and two-dimensional Poisson equation. For two-dimensional problems, both the families of Lagrange elements and Raviart-Thomas elements have been considered on uniform triangular and rectangular meshes. Numerical experiments reveal that many superconvergence properties of the standard Galerkin method are preserved by the least-squares finite element method.

A sparse algebraic multigrid method is studied as a cheap and accurate way to compute approximations of Schur complements of matrices arising from the discretization of some symmetric and positive definite partial differential operators. The construction of such a multigrid is discussed and numerical experiments are used to verify the properties of the method.

A sparse algebraic multigrid method is studied as a cheap and accurate way to compute approximations of Schur complements of matrices arising from the discretization of some symmetric and positive definite partial differential operators. The construction of such a multigrid is discussed and numerical experiments are used to verify the properties of the method.

Assuming an incident wave to be a field source, we calculate the field potential in a neighborhood of an inhomogeneous body. This problem which has been formulated in ${𝐑}^3$can be reduced to a bounded domain. Namely, a boundary condition for the potential is formulated on a sphere. Then the potential satisfies a well posed boundary value problem in a ball containing the body. A numerical approximation is suggested and its convergence is analyzed.

This contribution gives an overview of current research in applying object oriented programming to scientific computing at the computational mechanics laboratory (LABMEC) at the school of civil engineering – UNICAMP. The main goal of applying object oriented programming to scientific computing is to implement increasingly complex algorithms in a structured manner and to hide the complexity behind a simple user interface. The following areas are current topics of research and documented within the...

This contribution gives an overview of current research in applying object oriented programming to scientific computing at the computational mechanics laboratory (LABMEC) at the school of civil engineering – UNICAMP. The main goal of applying object oriented programming to scientific computing is to implement increasingly complex algorithms in a structured manner and to hide the complexity behind a simple user interface. The following areas are current topics of research and documented within the...

Over the last three decades Computational Fluid Dynamics (CFD) has gradually joined the wind tunnel and flight test as a primary flow analysis tool for aerodynamic designers. CFD has had its most favorable impact on the aerodynamic design of the high-speed cruise configuration of a transport. This success has raised expectations among aerodynamicists that the applicability of CFD can be extended to the full flight envelope. However, the complex nature...

This work is devoted to the study of a two-dimensional vector Poisson equation with the normal component of the unknown and the value of the divergence of the unknown prescribed simultaneously on the entire boundary. These two scalar boundary conditions appear prima facie alternative in a standard variational framework. An original variational formulation of this boundary value problem is proposed here. Furthermore, an uncoupled solution algorithm is introduced together with its finite element...