In this paper we present two versions of the central local discontinuous Galerkin (LDG) method on overlapping cells for solving diffusion equations, and provide their stability analysis and error estimates for the linear heat equation. A comparison between the traditional LDG method on a single mesh and the two versions of the central LDG method on overlapping cells is also made. Numerical experiments are provided to validate the quantitative conclusions from the analysis and to support conclusions...

In this paper we present two versions of the central local discontinuous Galerkin (LDG) method on overlapping cells for solving diffusion equations, and provide their stability analysis and error estimates for the linear heat equation. A comparison between the traditional LDG method on a single mesh and the two versions of the central LDG method on overlapping cells is also made. Numerical experiments are provided to validate the quantitative conclusions from the analysis and to support conclusions...

Let $T$ be a positive number or $+\infty$. We characterize all subsets $M$ of ${ℝ}^n\times ]0,T[$ such that $$\underset{X\in {ℝ}^n\times ]0,T[}{inf}u\left(X\right)=\underset{X\in M}{inf}u\left(X\right)i$$ for every positive parabolic function $u$ on ${ℝ}^n\times ]0,T[$ in terms of coparabolic (minimal) thinness of the set $M_{\delta }={\cup }_{(x,t)\in M}{B}^p((x,t),\delta t)$, where $\delta \in (0,1)$ and $B^p((x,t),r)$ is the “heat ball” with the “center” $(x,t)$ and radius $r$. Examples of different types of sets which can be used instead of “heat balls” are given. It is proved that (i) is equivalent to the condition ${sup}_{X\in {ℝ}^n\times {ℝ}^{+}}u\left(X\right)={sup}_{X\in M}u\left(X\right)$ for every bounded parabolic function on ${ℝ}^n\times {ℝ}^{+}$ and hence to all equivalent conditions given in the article [7]....

Abstract. We study a Neumann problem for the heat equation in a cylindrical domain with $C^1$-base and data in $h_c^1$, a subspace of $L$1. We derive our results, considering the action of an adjoint operator on $B_{T}MOC$, a predual of $h_c^1$, and using known properties of this last space.

Sono dati nuovi teoremi di esistenza per soluzioni regolari di equazioni di evoluzione paraboliche astratte con applicazioni all'equazione del calore in spazi di funzioni holderiane e alle equazioni semilineari.

European directives and related national technical standards force the substantial reduction of energy consumption of all types of buildings. This can be done thanks to the massive insulation and the improvement of quality of building enclosures, using the simple evaluation assuming the one-dimensional stationary heat conduction. However, recent applications of advanced materials, structures and technologies force the proper physical, mathematical and computational analysis coming from the thermodynamic...

We study boundary control problems for the wave, heat, and Schrödinger equations on a finite graph. We suppose that the graph is a tree (i.e., it does not contain cycles), and on each edge an equation is defined. The control is acting through the Dirichlet condition applied to all or all but one boundary vertices. Exact controllability in L₂-classes of controls is proved and sharp estimates of the time of controllability are obtained for the wave equation. Null controllability for the heat equation...

We discuss several new results on nonnegative approximate controllability for the one-dimensional Heat equation governed by either multiplicative or nonnegative additive control, acting within a proper subset of the space domain at every moment of time. Our methods allow us to link these two types of controls to some extend. The main results include approximate controllability properties both for the static and mobile control supports.

We show that a harmonic function which vanishes continuously on an open set of the boundary of a convex domain cannot have a normal derivative which vanishes on a subset of positive surface measure. We also prove a similar result for caloric functions vanishing on the lateral boundary of a convex cylinder.