### An application of the finite volume method to the bio-heat-transfer-equation in premature infants.

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The aim of this paper is to present how to make a dedicaded computed language polymorphic and multi type, in C++ to solve partial differential equations with the finite element method. The driving idea is to make the language as close as possible to the mathematical notation.

The aim of this paper is to present how to make a dedicaded computed language polymorphic and multi type, in C++to solve partial differential equations with the finite element method. The driving idea is to make the language as close as possible to the mathematical notation.

In this article, we present a numerical scheme based on a finite element method in order to solve a time-dependent convection-diffusion equation problem and satisfy some conservation properties. In particular, our scheme is able to conserve the total energy for a heat equation or the total mass of a solute in a fluid for a concentration equation, even if the approximation of the velocity field is not completely divergence-free. We establish a priori errror estimates for this scheme and we give some...

In this paper we derive a priori error estimates for linear-quadratic elliptic optimal control problems with finite dimensional control space and state constraints in the whole domain, which can be written as semi-infinite optimization problems. Numerical experiments are conducted to ilustrate our theory.

We consider a fully practical finite element approximation of the following degenerate system $$\frac{\partial}{\partial t}\rho \left(u\right)-\nabla .\left(\phantom{\rule{0.166667em}{0ex}}\alpha \left(u\right)\phantom{\rule{0.166667em}{0ex}}\nabla u\right)\ni \sigma \left(u\right)\phantom{\rule{0.166667em}{0ex}}{\left|\nabla \phi \right|}^{2},\phantom{\rule{1.0em}{0ex}}\nabla .\left(\phantom{\rule{0.166667em}{0ex}}\sigma \left(u\right)\phantom{\rule{0.166667em}{0ex}}\nabla \phi \right)=0$$ subject to an initial condition on the temperature, u, and boundary conditions on both u and the electric potential, ϕ. In the above p(u) is the enthalpy incorporating the latent heat of melting, α(u) > 0 is the temperature dependent heat conductivity, and σ(u) > 0 is the electrical conductivity. The latter is zero in the frozen zone, u ≤ 0, which gives rise to the degeneracy in this Stefan...

We consider a fully practical finite element approximation of the following degenerate system$$\phantom{\rule{-56.9055pt}{0ex}}\frac{\partial}{\partial t}\rho \left(u\right)-\nabla .\left(\phantom{\rule{0.166667em}{0ex}}\alpha \left(u\right)\phantom{\rule{0.166667em}{0ex}}\nabla u\right)\ni \sigma \left(u\right)\phantom{\rule{0.166667em}{0ex}}{\left|\nabla \phi \right|}^{2},\phantom{\rule{1.0em}{0ex}}\nabla .\left(\phantom{\rule{0.166667em}{0ex}}\sigma \left(u\right)\phantom{\rule{0.166667em}{0ex}}\nabla \phi \right)=0$$subject to an initial condition on the temperature, $u$, and boundary conditions on both $u$ and the electric potential, $\phi $. In the above $\rho \left(u\right)$ is the enthalpy incorporating the latent heat of melting, $\alpha \left(u\right)\>0$ is the temperature dependent heat conductivity, and $\sigma \left(u\right)\ge 0$ is the electrical conductivity. The latter is zero in the frozen zone, $u\le 0$, which gives rise to the degeneracy in this Stefan system. In addition to showing stability...

We analyze two numerical schemes of Euler type in time and C0 finite-element type with ${\mathbb{P}}_{1}$-approximation in space for solving a phase-field model of a binary alloy with thermal properties. This model is written as a highly non-linear parabolic system with three unknowns: phase-field, solute concentration and temperature, where the diffusion for the temperature and solute concentration may degenerate. The first scheme is nonlinear, unconditionally stable and convergent. The other scheme is linear...