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A growing body of literature has been leveraging techniques of machine learning (ML) to build novel approaches to approximating the solutions to partial differential equations. Noticeably absent from the literature is a systematic…
In this article, a numerical scheme to find approximate solutions to the McKendrick-Von Foerster equation with diffusion (M-V-D) is presented. The main difficulty in employing the standard analysis to study the properties of this scheme is…
In this research note we provide a variational basis for the optimal artificial diffusion method, which has been a cornerstone in developing many stabilized methods. The optimal artificial diffusion method produces exact nodal solutions…
We propose an energy-stable parametric finite element method (ES-PFEM) for simulating solid-state dewetting of thin films in two dimensions via a sharp-interface model, which is governed by surface diffusion and contact line (point)…
We are interested in the numerical solution of nonsymmetric linear systems arising from the discretization of convection-diffusion partial differential equations with separable coefficients and dominant convection. Preconditioners based on…
In this paper we present a new Eulerian finite element method for the discretization of scalar partial differential equations on evolving surfaces. In this method we use the restriction of standard space-time finite element spaces on a…
An implicit finite difference method with non-uniform timesteps for solving the fractional diffusion equation in the Caputo form is proposed. The method allows one to build adaptive methods where the size of the timesteps is adjusted to the…
As model problem we consider the prototype for flow and transport of a concentration in porous media in an interior domain and couple it with a diffusion process in the corresponding unbounded exterior domain. To solve the problem we…
In this paper, we consider separating the discretisation of the diffusive and advective fluxes in the complete flux scheme. This allows the combination of several discretisation methods for the homogeneous flux with the complete flux (CF)…
The Reynolds equation, combined with the Elrod algorithm for including the effect of cavitation, resembles a nonlinear convection-diffusion-reaction (CDR) equation. Its solution by finite elements is prone to oscillations in…
The quasidiffusion (QD) method, also known as the Variable Eddington Factor (VEF) method in the astrophysical community, is an established iterative method for accelerating source iterations in SN calculations. A great advantage of the QD…
We propose a structure-preserving parametric finite element method (SP-PFEM) for discretizing the surface diffusion of a closed curve in two dimensions (2D) or surface in three dimensions (3D). Here the "structure-preserving" refers to…
We present a new stabilization technique for multiscale convection diffusion problems. Stabilization for these problems has been a challenging task, especially for the case with high Peclet numbers. Our method is based on a constraint…
A high-order finite element method is proposed to solve the nonlinear convection-diffusion equation on a time-varying domain whose boundary is implicitly driven by the solution of the equation. The method is semi-implicit in the sense that…
In this paper we establish a stability barrier of a class of high-order Hermite-type discretization of 1D advection equations underlying the hybrid-variable (HV) and active flux (AF) methods. These methods seek numerical approximations to…
We consider systematic numerical approximation of a viscoelastic phase separation model that describes the demixing of a polymer solvent mixture. An unconditionally stable discretisation method is proposed based on a finite element…
We present a full space-time numerical solution of the advection-diffusion equation using a continuous Galerkin finite element method on conforming meshes. The Galerkin/least-square method is employed to ensure stability of the discrete…
The numerical solution of a nonlinear and space-fractional anti-diffusive equation used to model dune morphodynamics is considered. Spatial discretization is effected using a finite element method whereas the Crank-Nicolson scheme is used…
We numerically investigate the possibility of defining stabilization-free Virtual Element (VEM) discretizations of advection-diffusion problems in the advection-dominated regime. To this end, we consider a SUPG stabilized formulation of the…
A semidiscrete Galerkin finite element method applied to time-fractional diffusion equations with time-space dependent diffusivity on bounded convex spatial domains will be studied. The main focus is on achieving optimal error results with…