Marián Slodička

A Robust and Efficient Linearization Scheme for Doubly Nonlinear and Degenerate Parabolic Problems Arising in Flow in Porous Media

We study a nonlinear degenerate convection-diffusion model problem having an application in groundwater aquifer and petroleum reservoir simulation. The true solution typically possesses low regularity, and therefore special numerical techniques for its approximation are needed. We design a robust, efficient, and reliable linear relaxation approximation scheme. We prove the convergence of iterations at each time step in the

An analysis of inverse source problems with final time measured output data for the heat conduction equation: A semigroup approach

H^1(\Omega)

Determination of a time-dependent heat transfer coefficient in a nonlinear inverse heat conduction problem

-norm. Finally, the convergence of the approximate solution in corresponding functional spaces to its exact counterpart for the parabolic problem is shown.

Determination of a time-dependent heat transfer coefficient from non-standard boundary measurements

Abstract This paper presents a semigroup approach for inverse source problems for the abstract heat equation u t = A u + F , when the measured output data is given in the form the final overdetermination u T ( x ) ≔ u ( x , T ) . A representation formula for a solution of the inverse source problem is proposed. This representation shows a non-uniqueness structure of the inverse problem solution, and also permits one to derive a sufficient condition for uniqueness. Some examples related to identifying the unknown spacewise and time-dependent heat sources f ( x ) and h ( t ) of the heat equation u t = u x x + f ( x ) h ( t ) , from the final overdetermination or from a single point time measurement are presented.

A numerical scheme for a Maxwell-Landau-Lifshitz-Gilbert system

In this article, the determination of the time-dependent heat transfer coefficient, involving nonlinear boundary conditions of the third kind in the one-dimensional transient heat conduction from a non-standard boundary measurement is investigated. For this inverse, nonlinear, ill-posed problem, the existence and uniqueness of the solution are proved. Numerical results are obtained, using the boundary element method, and discussed.

Error analysis in the reconstruction of a convolution kernel in a semilinear parabolic problem with integral overdetermination

In this paper the determination of the time-dependent heat transfer coefficient in one-dimensional transient heat conduction from a non-standard boundary measurement is investigated. For this inverse nonlinear ill-posed problem the uniqueness of the solution holds. Numerical results obtained using the boundary element method are presented and discussed.

Determination of a Robin coefficient in semilinear parabolic problems by means of boundary measurements

We consider Maxwells equations together with a nonlinear dissipative magnetic law described by the Landau-Lifshitz-Gilbert equation. The paper is devoted to the numerical study of this problem. We introduce two algorithms for the time discretization. The modulus of magnetization is conserved in both cases. Assuming a sufficiently smooth electromagnetic field, we derive the error estimates for both approximation schemes.

Determination of a solely time-dependent source in a semilinear parabolic problem by means of boundary measurements

A semilinear parabolic problem of second order with an unknown solely time-dependent convolution kernel is considered. An additional given global measurement (a space integral of the solution) ensures the existence of a unique weak solution. The unknown kernel function can be approximated by a time-discrete numerical scheme based on Backward Eulers method (Rothes method). In this contribution, an error analysis for the time discretization is performed of the existing numerical algorithm. Numerical experiments support the theoretically obtained results.

Recovery of a space-dependent vector source in thermoelastic systems

We consider a semilinear parabolic initial boundary value problem of second order in a bounded domain Ω ⊂ N with a Lipschitz continuous boundary. We study the problem of determination of the Robin coefficient at the boundary part Γnon from a nonlocal boundary condition. We prove the well posedness of the problem. We design a numerical scheme for its approximation and perform the error analysis.

A numerical approach for the determination of a missing boundary data in elliptic problems

A semilinear parabolic problem of second order with an unknown solely time-dependent source term is studied. The missing source is recovered from an additional integral measurement over the boundary. The global in time existence, uniqueness as well as the regularity of a solution are addressed. A new numerical scheme based on Rothes method is designed and convergence of iterates towards the exact solution is shown.