# Mathematics seminar series 2015/2016

## Mathematics Research Seminar Programme 2015-16

#### Thursday 17th March 2016, 15.05-16.05, Room: TSU102 (Thornton Science Park)

Dr. Dumutru Trucu, University of Dundee.

"New horizons in multiscale modelling and analysis: the novel concept of three-scale convergence"

ABSTRACT: In this work we propose a new notion of multiscale convergence, called "three-scale", which aims to give a topological framework in which to assess complex processes occurring at three different scales or levels within a heterogeneous medium.
While these kind of three-scale processes occur in many physical, chemical and biological situations, this work was motivated by our specific research interest that is focused on the development of a novel multiscale modelling framework for cancer cells invasion of human body tissue. The genuinely multiscale nature of the cancer invasion process is explored in this new framework via a novel three-scale moving boundary modelling approach. Naturally, this modelling leads to a question concerning the establishment of a fundamental framework that would enable a rigorous analysis of involved operators.
The new multiscale analysis concept that we introduce here generalises and extends the notion of two-scale convergence, a well-established concept that is commonly used for obtaining an averaged, asymptotic value (homogenization) of processes that exist on two different spatial scales. The well-posedness of this new concept is justified via a compactness theorem which ensures that all bounded sequences of square integrable functions over a bounded domain are relative compact with respect to the three-scale convergence. This is taken further by giving a boundedness characterisation of three-scale convergent sequences and is then continued with the introduction of the notion of "strong three-scale convergence" whose well-posedness is also discussed. Finally, the three-scale convergence of the gradients is established.

#### Thursday 10th March 2016, 15.05-16.05, Room: TSU102 (Thornton Science Park)

Dr Athanasios A. Pantelous, University of Liverpool.

"Stochastic response determination of linear and nonlinear dynamical systems with singular matrices"

ABSTRACT: A generalization of random vibration techniques is developed for determining the stochastic response of linear and nonlinear dynamical systems with singular matrices. This system modeling can arise when a greater than the minimum number of coordinates is utilized, and can be advantageous, for instance, in cases of complex multi-body systems where the explicit formulation of the equations of motion can be a nontrivial task. In such cases, the introduction of additional/redundant degrees of freedom can facilitate the formulation of the equations of motion in a less labour intensive manner. First, relying on the generalized matrix inverse theory and on the Moore-Penrose (M-P) matrix inverse, standard concepts, relationships, and equations of the linear random vibration theory are extended and generalized herein to account for systems with singular matrices. It has been shown that adopting a state-variable formulation, equations governing the system response mean vector and covariance matrix are formed and solved. In this paper, an alternative frequency domain approach for stochastic response determination of linear systems with singular matrices is developed as well. Further, the above solution framework is generalized to account for nonlinear systems. To this aim, the potent statistical linearization technique is generalized to account for systems with singular matrices.

#### Prof. Guang Zhang, School of Science, Tianjin University of Commerce, China.

"Existence of positive solutions for a class of nonlinear algebraic systems"

ABSTRACT: In this talk, I will introduce some existence results of positive solutions for a class of nonlinear algebraic systems which have been shown to have interesting applications in various areas such as difference equations, boundary value problems, dynamical networks, existence of periodic solutions, stochastic processes, numerical analysis, etc.

#### Thursday 28th January 2016, 15.05-16.05, Room: TSU102 (Thornton Science Park)

Dr Zhiqiang Li (Lvling University) and Dr Yubin Yan (University of Chester)

"Error estimates of a high order numerical method for solving linear fractional differential equations"

ABSTRACT: In this talk, we first introduce an alternative proof of the error estimates of the numerical methods for solving linear fractional differential equations proposed in Diethelm, (1997),  where a first-degree compound quadrature formula was used to approximate the Hadamard finite-part integral and the convergence order of the proposed numerical method is $O(\Delta t^{2-\alpha}), 0 < \alpha <1$, where $\alpha$ is the order of the fractional derivative and $\Delta t$ is the step size. We then use the similar idea to prove the error estimates of a high order numerical method for solving linear fractional differential equations proposed in Yan, Pal and Ford (2013), where a second-degree compound quadrature formula was used to approximate the Hadamard finite-part integral and we show that the convergence order of the numerical method is $O(\Delta t^{3-\alpha}), 0< \alpha <1$. The numerical examples are given to show that the numerical results are consistent with the theoretical results."

#### Thursday 21st January  2016, 15.10-16.10, Room: TSU102 (Thornton Science Park)

Dr Gabriel Barrenechea (University of Stratchclyde)

"Certified upper and lower bounds for the eigenvalues of the Maxwell operator."

ABSTRACT:  We propose a strategy which allows computing eigenvalue enclosures for the  Maxwell operator by means of the  finite element method. The origins of this strategy can be traced back to over 20 years ago. One of its main features lies in the fact that it can be implemented on any type of regular mesh (structured or otherwise) and any type of elements  (nodal or otherwise). In the first part of the talk, we formulate a general framework which is free from spectral pollution and allows estimation of eigenfunctions.  We then prove the convergence of the method, which implies precise convergence rates for nodal finite elements. Various numerical experiments on benchmark geometries, with and without symmetries, are reported.

#### Thursday 3rd December 2015 15.05-16.05, Room: TSU102 (Thornton Science Park)

Dr Andrea Cangiani, University of Leicester

"Numerical PDEs on general meshes"

ABSTRACT: Real-life models are often characterised by localised features. For instance, solution layers/singularities, domains with complicated/moving boundaries, and multi-physics matching. These features make the design of accurate numerical solutions challenging, or even out of reach unless computational resources are smartly allocated. Complexity reduction can be achieved through the following framework: new Finite Element Method (FEM) approaches allowing for general partitioning of the computational domain combined with automatic adaptive meshing. For instance, general non-matching/curved mesh elements can ease the treatment of multi-physics boundaries, while local geometric and solution features can be resolved adaptively. I will present two approaches to extending the FEM to general meshes while maintaining the ease of implementation and computational cost comparable to that of standard FEM:  the Virtual Element Method (VEM) and a discontinuous Galerkin method. Recent work on adaptive algorithms based on rigorous a posteriori error bounds will also be presented and demonstrated on problems with internal possibly curved interfaces modelling semi-permeable membranes.

#### Thursday 29th October 2015, 15.10-16.10, Room: TSU102 (Thornton Science Park)

Dr Fotini Karakatsani (University of Chester)

"The effect of mesh modification in time on the error control of fully discrete approximations of evolution equations".

ABSTRACT: We consider fully discrete schemes for linear parabolic problems discretised by the Crank–Nicolson method in time and the standard finite element method in space. We study the effect of mesh modification on the stability of fully discrete approximations as well as its influence on residual-based a posteriori error estimators. We focus mainly on the qualitative, analytical and computational behavior of the schemes and the error estimators.

#### Fri 2 October 2015, 11.30-12.30, Room: TSU102 (Thornton Science Park)

Prof. Andreas Prohl (University of Tuebingen, Germany)

"Strong convergence with rates for discretizations of SPDEs with nonlinear drift"

ABSTRACT: I discuss the convergence analysis for space-time discretizations of three nonlinear SPDE's: the stochastic Navier-Stokes equation, the stochastic Allen-Cahn equation, and the stochastic mean curvature flow of planar curves of graphs. Depending on the drift operator, optimal rates w.r.t. strong convergence are valid for errors on large subsets, or on the whole sample set.