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...

A priori estimates and strong solvability results in Sobolev space $W^{2,p}\left(\Omega \right)$, $1<p<\infty$ are proved for the regular oblique derivative problem $$\left\{\begin{array}{c}\sum _{i,j=1}^n{a}^{ij}\left(x\right)\frac{{\partial }^{2}u}{\partial x_i\partial x_j}=f\left(x\right)\text{a.e.}\Omega \hfill \\ \frac{\partial u}{\partial \ell }+\sigma \left(x\right)u=\varphi \left(x\right)\text{on}\partial \Omega \hfill \end{array}\right.$$ when the principal coefficients $a^{ij}$ are $VMO\cap L^{\infty }$ functions.

In this paper, we make some observations on the work of Di Fazio concerning $W^{1,p}$ estimates, $1<p<\mathrm{\infty }$, for solutions of elliptic equations $\text{div}A\nabla u=\text{div}f$ , on a domain $\mathrm{\Omega }$ with Dirichlet data $0$ whenever $A\in VMO\left(\mathrm{\Omega }\right)$ and $f\in L^p\left(\mathrm{\Omega }\right)$. We weaken the assumptions allowing real and complex non-symmetric operators and $C^1$ boundary. We also consider the corresponding inhomogeneous Neumann problem for which we prove the similar result. The main tool is an appropriate representation for the Green (and Neumann) function on the upper half space. We propose...

A Bernoulli free boundary problem with geometrical constraints is studied. The domain Ω is constrained to lie in the half space determined by x1 ≥ 0 and its boundary to contain a segment of the hyperplane  {x1 = 0}  where non-homogeneous Dirichlet conditions are imposed. We are then looking for the solution of a partial differential equation satisfying a Dirichlet and a Neumann boundary condition simultaneously on the free boundary. The existence and uniqueness of a solution have already been addressed...

A Bernoulli free boundary problem with geometrical constraints is studied. The domain Ω is constrained to lie in the half space determined by x1 ≥ 0 and its boundary to contain a segment of the hyperplane  {x1 = 0}  where non-homogeneous Dirichlet conditions are imposed. We are then looking for the solution of a partial differential equation satisfying a Dirichlet and a Neumann boundary condition simultaneously on the free boundary. The existence and uniqueness of a solution have already been addressed...

We prove some comparison results for Monge-Ampère type equations in dimension two. We consider also the case of eigenfunctions and we prove a kind of reverse inequalities.

We discuss a parallel implementation of the domain decomposition method based on the macro-hybrid formulation of a second order elliptic equation and on an approximation by the mortar element method. The discretization leads to an algebraic saddle- point problem. An iterative method with a block- diagonal preconditioner is used for solving the saddle- point problem. A parallel implementation of the method is emphasized. Finally the results of numerical experiments are presented.

We obtain a description of the spectrum and estimates for generalized positive solutions of -Δu = λ(f(x) + h(u)) in Ω, ${u|}_{\partial \Omega }=0$, where f(x) and h(u) satisfy minimal regularity assumptions.