Let $AV\to V^{\text{'}}$ be a strongly elliptic operator on a $d$-dimensional manifold $D$ (polyhedra or boundaries of polyhedra are also allowed). An operator equation $Au=f$ with stochastic data $f$ is considered. The goal of the computation is the mean field and higher moments ${ℳ}^{1}u\in V$, ${ℳ}^{2}u\in V\otimes V$, $...$, ${ℳ}^{k}u\in V\otimes \cdots \otimes V$ of the solution. We discretize the mean field problem using a FEM with hierarchical basis and $N$ degrees of freedom. We present a Monte-Carlo algorithm and a deterministic algorithm for the approximation of the moment ${ℳ}^{k}u$ for $k\ge 1$....

Let $L$ be a parabolic second order differential operator on the domain $\overline{\Pi }=\left[0,T\right]\times ℝ.$ Given a function $\widehat{u}:ℝ\to R$ and $\widehat{x}\>0$ such that the support of $\widehat{u}$ is contained in $(-\infty ,-\widehat{x}]$, we let $\widehat{y}:\overline{\Pi }\to ℝ$ be the solution to the equation: $$L\widehat{y}=0,\widehat{y}{|}_{\left\{0\right\}\times ℝ}=\widehat{u}.$$ Given positive bounds $0\<x_0\<x_1,$ we seek a function $u$ with support in $\left[x_0,x_1\right]$ such that the corresponding solution $y$ satisfies: $$y(t,0)=\widehat{y}(t,0)\forall t\in \left[0,T\right].$$ We prove in this article that, under some regularity conditions on the coefficients of $L,$ continuous solutions are unique and dense in the...

An energy analysis is carried out for the usual semidiscrete Galerkin method for the semilinear equation in the region$\Omega$ (E) ${\left(t{u}_t\right)}_t={\sum }_{i,j=1}{\left(a_{ij}\left(x\right)u_{x_i}\right)}_{x_j}-a_0\left(x\right)u+f\left(u\right)$, subject to the initial and boundary conditions, $u=0$ on $\partial \Omega$ and $u(x,0)=u_0$. (E) is degenerate at $t=0$ and thus, even in the case $f\equiv 0$, time derivatives of $u$ will blow up as $t\to 0$. Also, in the case where $f$ is locally Lipschitz, solutions of (E) can blow up for $t>0$ in finite time. Stability and convergence of the scheme in $W^{2,1}$ is shown in the linear case without assuming $u_{tt}$ (which can blow...

We study the asymptotic behavior of a semi-discrete numerical approximation for a pair of heat equations $u_t=\Delta u$, $v_t=\Delta v$ in $\Omega \times (0,T)$; fully coupled by the boundary conditions $\frac{\partial u}{\partial \eta }=u^{p_{11}}{v}^{p_{12}}$, $\frac{\partial v}{\partial \eta }=u^{p_{21}}{v}^{p_{22}}$ on $\partial \Omega \times (0,T)$, where $\Omega$ is a bounded smooth domain in ${ℝ}^d$. We focus in the existence or not of non-simultaneous blow-up for a semi-discrete approximation $(U,V)$. We prove that if $U$ blows up in finite time then $V$ can fail to blow up if and only if $p_{11}\>1$ and $p_{21}\<2(p_{11}-1)$, which is the same condition as the one for non-simultaneous blow-up in the continuous problem....

We present a completely new $hp$-anisotropic mesh adaptation technique for the numerical solution of partial differential equations with the aid of a discontinuous piecewise polynomial approximation. This approach generates general anisotropic triangular grids and the corresponding degrees of polynomial approximation based on the minimization of the interpolation error. We develop the theoretical background of this approach and present a numerical example demonstrating the efficiency of...

Let $y\left(h\right)(t,x)$ be one solution to $${\partial }_{t}y(t,x)-\sum _{i,j=1}^n{\partial }_j\left(a_{ij}\left(x\right){\partial }_{i}y(t,x)\right)=h(t,x),0\<t\<T,x\in \Omega$$ with a non-homogeneous term $h$, and ${y|}_{(0,T)\times \partial \Omega }=0$, where $\Omega \subset {ℝ}^n$ is a bounded domain. We discuss an inverse problem of determining $n(n+1)/2$ unknown functions $a_{ij}$ by $\{{\partial }_{\nu }y\left(h_{\ell }\right){|}_{(0,T)\times {\Gamma }_0}$, $y\left(h_{\ell }\right)(\theta ,·){\}}_{1\le \ell \le {\ell }_0}$ after selecting input sources $h_1,...,h_{{\ell }_0}$ suitably, where ${\Gamma }_0$ is an arbitrary subboundary, ${\partial }_{\nu }$ denotes the normal derivative, $0\<\theta \<T$ and ${\ell }_0\in \mathbb{N}$. In the case of ${\ell }_0={(n+1)}^{2}n/2$, we prove the Lipschitz stability in the inverse problem if we choose $(h_1,...,h_{{\ell }_0})$ from a set $\mathscr{H}\subset {\left\{C_0^{\infty }((0,T)\times \omega )\right\}}^{{\ell }_0}$ with an arbitrarily fixed subdomain $\omega \subset \Omega$. Moreover we can take ${\ell }_0=(n+3)n/2$ by making...

The initial boundary value problem for a beam is considered in the Timoshenko model. Assuming the analyticity of the initial conditions, it is proved that the problem is solvable throughout the time interval. After that, a numerical algorithm, consisting of three steps, is constructed. The solution is approximated with respect to the spatial and time variables using the Galerkin method and a Crank–Nicholson type scheme. The system of equations obtained by discretization is solved by...

We develop a new technique which, for the given smooth function, generates the anisotropic triangular grid and the corresponding polynomial approximation degrees based on the minimization of the interpolation error in the broken $H^1$-seminorm. This technique can be employed for the numerical solution of boundary value problems with the aid of finite element methods. We present the theoretical background of this approach and show several numerical examples demonstrating the efficiency of...