A Posteriori Error Estimates for Nonconforming Approximations of Evolutionary Convection-Diffusion Problems

We derive computable upper bounds for the difference between an exact solution of the evolutionary convection-diffusion problem and an approximation of this solution. The estimates are obtained by certain transformations of the integral identity that defines the generalized solution. These estimates depend on neither special properties of the exact solution nor its approximation, and involve only global constants coming from embedding inequalities. The estimates are first derived for functions in the corresponding energy space, and then possible extensions to classes of piecewise continuous approximations are discussed.Comment: 10 page

PDEs in Moving Time Dependent Domains

In this work we study partial differential equations defined in a domain that moves in time according to the flow of a given ordinary differential equation, starting out of a given initial domain. We first derive a formulation for a particular case of partial differential equations known as balance equations. For this kind of equations we find the equivalent partial differential equations in the initial domain and later we study some particular cases with and without diffusion. We also analyze general second order differential equations, not necessarily of balance type. The equations without diffusion are solved using the characteristics method. We also prove that the diffusion equations, endowed with Dirichlet boundary conditions and initial data, are well posed in the moving domain. For this we show that the principal part of the equivalent equation in the initial domain is uniformly elliptic. We then prove a version of the weak maximum principle for an equation in a moving domain. Finally we perform suitable energy estimates in the moving domain and give sufficient conditions for the solution to converge to zero as time goes to infinity.Comment: pp 559-577. Without Bounds: A Scientific Canvas of Nonlinearity and Complex Dynamics (2013) p. 36

Asymptotics for models of non-stationary diffusion in domains with a surface distribution of obstacles

We consider a time-dependent model for the diffusion of a substance through an incompressible fluid in a perforated domain ??, urn:x-wiley:mma:media:mma5323:mma5323-math-0001 with n?=?3,4. The fluid flows in a domain containing a periodical set of ?obstacles? (?\??) placed along an inner (n???1)?dimensional manifold urn:x-wiley:mma:media:mma5323:mma5323-math-0002. The size of the obstacles is much smaller than the size of the characteristic period ?. An advection term appears in the partial differential equation linking the fluid velocity with the concentration, while we assume a nonlinear adsorption law on the boundary of the obstacles. This law involves a monotone nonlinear function ? of the concentration and a large adsorption parameter. The ?critical adsorption parameter? depends on the size of the obstacles , and, for different sizes, we derive the time?dependent homogenized models. These models contain a ?strange term? in the transmission conditions on ?, which is a nonlinear function and inherits the properties of ?. The case in which the fluid velocity and the concentration do not interact is also considered for n???3.The authors would like to thank the anonymous referees for their careful reading of the manupscript and useful comments. The work has been partially supported by MINECO, MTM2013-44883-P

Local regularity for fractional heat equations

We prove the maximal local regularity of weak solutions to the parabolic problem associated with the fractional Laplacian with homogeneous Dirichlet boundary conditions on an arbitrary bounded open set $\Omega\subset\mathbb{R}^N$. Proofs combine classical abstract regularity results for parabolic equations with some new local regularity results for the associated elliptic problems.Comment: arXiv admin note: substantial text overlap with arXiv:1704.0756

Numerical studies of the Lagrangian approach for reconstruction of the conductivity in a waveguide

We consider an inverse problem of reconstructing the conductivity function in a hyperbolic equation using single space-time domain noisy observations of the solution on the backscattering boundary of the computational domain. We formulate our inverse problem as an optimization problem and use Lagrangian approach to minimize the corresponding Tikhonov functional. We present a theorem of a local strong convexity of our functional and derive error estimates between computed and regularized as well as exact solutions of this functional, correspondingly. In numerical simulations we apply domain decomposition finite element-finite difference method for minimization of the Lagrangian. Our computational study shows efficiency of the proposed method in the reconstruction of the conductivity function in three dimensions