Search results for "elliptic partial differential equations"
showing 2 items of 2 documents
Regularity of renormalized solutions to nonlinear elliptic equations away from the support of measure data
2018
We prove boundedness and continuity for solutions to the Dirichlet problem for the equation $$ - {\rm{div}}(a(x,\nabla u)) = h(x,u) + \mu ,\;\;\;\;\;{\rm{in}}\;{\rm{\Omega }} \subset \mathbb{R}^{N},$$ where the left-hand side is a Leray-Lions operator from $$- {W}^{1,p}_0(\Omega)$$ into W−1,p′(Ω) with 1 < p < N, h(x,s) is a Caratheodory function which grows like ∣s∣p−1 and μ is a finite Radon measure. We prove that renormalized solutions, though not globally bounded, are Holder-continuous far from the support of μ.
On the a posteriori error analysis for linear Fokker-Planck models in convection-dominated diffusion problems
2018
This work is aimed at the derivation of reliable and efficient a posteriori error estimates for convection-dominated diffusion problems motivated by a linear Fokker-Planck problem appearing in computational neuroscience. We obtain computable error bounds of the functional type for the static and time-dependent case and for different boundary conditions (mixed and pure Neumann boundary conditions). Finally, we present a set of various numerical examples including discussions on mesh adaptivity and space-time discretisation. The numerical results confirm the reliability and efficiency of the error estimates derived.