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The problem of robust nonlinear energy-to-peak filtering for nonlinear descriptor systems with model uncertainties is addressed. The system is assumed to have nonlinearities both in the state and output equations as well as norm-bounded…
Building on the successes of local kernel methods for approximating the solutions to partial differential equations (PDE) and the evaluation of definite integrals (quadrature/cubature), a local estimate of the error in such approximations…
In this thesis we investigate how the nonlocalities affect the study of different PDEs coming from physics, and we analyze these equations under almost optimal assumptions of the nonlinearity. In particular, we focus on the fractional…
We study the continuity of weak solutions for quasilinear elliptic systems with source terms of critical growth arising from a transport-energy structure. The latter occurs frequently in connection with the first balance principles of…
Partial differential equations (PDEs) involving high contrast and oscillating coefficients are common in scientific and industrial applications. Numerical approximation of these PDEs is a challenging task that can be addressed, for example,…
The purpose of this work is the study of solution techniques for problems involving fractional powers of symmetric coercive elliptic operators in a bounded domain with Dirichlet boundary conditions. These operators can be realized as the…
In this paper we propose and analyze a new Multiscale Method for solving semi-linear elliptic problems with heterogeneous and highly variable coefficient functions. For this purpose we construct a generalized finite element basis that spans…
In this paper, we study a class of quasilinear elliptic equations involving both local and nonlocal operators with variable exponents. The problem exhibits singular nonlinearities along with a subcritical superlinear growth term and a…
In this article we study adaptive finite element methods (AFEM) with inexact solvers for a class of semilinear elliptic interface problems. We are particularly interested in nonlinear problems with discontinuous diffusion coefficients, such…
In power systems, large-scale optimisation problems are extensively used to plan for capacity expansion at the supra-national level. However, their cost-optimal solutions are often not exploitable by decision-makers who are preferably…
This paper studies the derivation of the quadratic porous medium equation and a class of cross-diffusion systems from nonlocal interactions. We prove convergence of solutions of a nonlocal interaction equation, resp. system, to solutions of…
Motivated by recent work on approximation of diffusion equations by deterministic interacting particle systems, we develop a nonlocal approximation for a range of linear and nonlinear diffusion equations and prove convergence of the method…
We propose, analyze, and test new iterative solvers for large-scale systems of linear algebraic equations arising from the finite element discretization of reduced optimality systems defining the finite element approximations to the…
In this work we present a generic framework for non-conforming finite elements on polytopal meshes, characterised by elements that can be generic polygons/polyhedra. We first present the functional framework on the example of a linear…
We propose and analyze an unfitted finite element method for solving elliptic problems on domains with curved boundaries and interfaces. The approximation space on the whole domain is obtained by the direct extension of the finite element…
We derive new boundary conditions and implementation procedures for nonlinear initial boundary value problems that lead to energy and entropy bounded solutions. A step-by-step procedure for general nonlinear hyperbolic problems on…
We develop subrepresentation inequalities for infinitely degenerate metrics, and obtain corresponding Poincare and Sobolev inequalities. We then derive conditions on the degenerate metric under which weak solutions to associated infinitely…
We have constructed new formulae for generation of solutions for the nonlinear heat equation and for the Burgers equation that are based on linearizing nonlocal transformations and on nonlocal symmetries of linear equations. Found nonlocal…
In many applications of practical interest, solutions of partial differential equation models arise as critical points of an underlying (energy) functional. If such solutions are saddle points, rather than being maxima or minima, then the…
Local linear instability refers to the linearized discrete operator exhibiting perturbation growth exceeding that of the corresponding continuous linearized problem. In the context of nonlinear entropy-stable discretizations, we argue that…