In this paper, we propose a method for the approximation of the solution of high-dimensional weakly coercive problems formulated in tensor spaces using low-rank approximation formats. The method can be seen as a perturbation of a minimal residual method with a measure of the residual corresponding to the error in a specified solution norm. The residual norm can be designed such that the resulting low-rank approximations are optimal with respect to particular norms of interest, thus allowing to take...

We consider a simple reaction-diffusion system exhibiting Turing's diffusion driven instability if supplemented with classical homogeneous mixed boundary conditions. We consider the case when the Neumann boundary condition is replaced by a unilateral condition of Signorini type on a part of the boundary and show the existence and location of bifurcation of stationary spatially non-homogeneous solutions. The nonsymmetric problem is reformulated as a single variational inequality with a potential...

Given a reaction-diffusion system which exhibits Turing's diffusion-driven instability, the influence of unilateral obstacles of opposite sign (source and sink) on bifurcation and critical points is studied. In particular, in some cases it is shown that spatially nonhomogeneous stationary solutions (spatial patterns) bifurcate from a basic spatially homogeneous steady state for an arbitrarily small ratio of diffusions of inhibitor and activator, while a sufficiently large ratio is necessary in the...

In this paper we propose and analyze stable variational formulations for convection diffusion problems starting from concepts introduced by Sangalli. We derive efficient and reliable a posteriori error estimators that are based on these formulations. The analysis of resulting adaptive solution concepts, when specialized to the setting suggested by Sangalli’s work, reveals partly unexpected phenomena related to the specific nature of the norms induced by the variational formulation. Several remedies,...

In this paper we propose and analyze stable variational formulations for convection diffusion problems starting from concepts introduced by Sangalli. We derive efficient and reliable a posteriori error estimators that are based on these formulations. The analysis of resulting adaptive solution concepts, when specialized to the setting suggested by Sangalli’s work, reveals partly unexpected phenomena related to the specific nature of the norms induced by the variational formulation. Several remedies,...

Let $W$ be a non-negative function of class ${\mathrm{C}}^3$ from ${ℝ}^2$ to $ℝ$, which vanishes exactly at two points $𝐚$ and $𝐛$. Let $S^1(𝐚,𝐛)$ be the set of functions of a real variable which tend to $𝐚$ at $-\infty$ and to $𝐛$ at $+\infty$ and whose one dimensional energy$$E_1\left(v\right)={\int }_{ℝ}\left[W\left(v\right)+|v^{\text{'}}{|}^2/2\right]\mathrm{d}x$$is finite. Assume that there exist two isolated minimizers $z_{+}$ and $z_{-}$ of the energy $E_1$ over $S^1(𝐚,𝐛)$. Under a mild coercivity condition on the potential $W$ and a generic spectral condition on the linearization of the one-dimensional Euler–Lagrange operator at $z_{+}$ and $z_{-}$, it is possible to prove...

Let W be a non-negative function of class C3 from ${}^2$ to $$, which vanishes exactly at two points a and b. Let S1(a, b) be the set of functions of a real variable which tend to a at -∞ and to b at +∞ and whose one dimensional energy $$E_1\left(v\right)={\int }_{\left[W\left(v\right)+{\left|v^{\text{'}}\right|}^2/2\right]}x$$ is finite. Assume that there exist two isolated minimizers z+ and z- of the energy E1 over S1(a, b). Under a mild coercivity condition on the potential W and a generic spectral condition on the linearization of the one-dimensional Euler–Lagrange operator at z+ and...