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In this article, we discuss the numerical solution of diffusion equations on random surfaces within the isogeometric framework. We describe in detail, how diffusion problems on random surfaces can be modelled and how quantities of interest…
A numerical method for approximating weak solutions of an aggregation equation with degenerate diffusion is introduced. The numerical method consists of a stabilized finite element method together with a mass lumping technique and an extra…
A Lagrangian-type numerical scheme called the "comoving mesh method" or CMM is developed for numerically solving certain classes of moving boundary problems which include, for example, the classical Hele-Shaw flow problem and the well-known…
We use a numerical-analytic technique to construct a sequence of successive approximations to the solution of a system of fractional differential equations, subject to Dirichlet boundary conditions. We prove the uniform convergence of the…
The dispersion process in particulate porous media at low saturation levels takes place over the surface elements of constituent particles and, as we have found previously by comparison with experiments, can be accurately described by…
We study numerical methods for the nonlinear partial differential equation that governs the motion of level sets by affine curvature. We show that standard finite difference schemes are nonlinearly unstable. We build convergent finite…
A finite difference method is constructed to solve singularly perturbed convection-diffusion problems posed on smooth domains. Constraints are imposed on the data so that only regular exponential boundary layers appear in the solution. A…
We consider a mechanical system with impact and n degrees of freedom, written in generalized coordinates. The system is not necessarily Lagrangian. The representative point of the system must remain inside a set of constraints K; the…
In this paper, we consider a numerical method to solve scattering problems with multi-periodic layers with different periodicities. The main tool applied in this paper is the Bloch transform. With this method, the problem is written into an…
We present a computational scheme that derives a global polynomial level set parametrisation for smooth closed surfaces from a regular surface-point set and prove its uniqueness. This enables us to approximate a broad class of smooth…
We solve the Plateau problem for marginally outer trapped surfaces in general Cauchy data sets. We employ the Perron method and tools from geometric measure theory to force and control a blow-up of Jang's equation. Substantial new geometric…
A numerical scheme is presented for solving the Helmholtz equation with Dirichlet or Neumann boundary conditions on piecewise smooth open curves, where the curves may have corners and multiple junctions. Existing integral equation methods…
We propose a novel multimesh rational approximation scheme for the numerical solution of the (homogeneous) Dirichlet problem for the spectral fractional Laplacian. The scheme combines a rational approximation of the function $\lambda…
A new method for numerical solving of boundary problem for ordinary differential equations with slowly varying coefficients which is aimed at better representation of solutions in the regions of their rapid oscillations or exponential…
A numerical scheme is presented to solve the one source near field refractor problem to arbitrary precision and it is proved that the scheme terminates in a finite number of iterations. The convergence of the algorithm depends upon proving…
This paper further develops the Method of Matched Sections (MMS), a robust numerical framework for the solution of boundary value problems governed by partial differential equations. It demonstrates its unique applicability to the…
In this article, we introduce and study three numerical methods for the Dirichlet Monge Amp\`ere equation in two dimensions. The approaches consist in considering new equivalent problems. The latter are discretized by a wide stencil finite…
We exhibit differential geometric structures that arise in numerical methods, based on the construction of Cauchy sequences, that are currently used to prove explicitly the existence of weak solutions to functional equations. We describe…
In this paper, we will study the Bloch transformed rough surface scattering problems, and propose a numerical method based on the Bloch transformed problems. Based on the mathematical theory of the scattering problems from locally perturbed…
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…