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In this work we derive equivalence relations between mimetic finite difference schemes on simplicial grids and modified N\'ed\'elec-Raviart-Thomas finite element methods for model problems in $\mathbf{H}(\operatorname{\mathbf{curl}})$ and…
We provide an approach to implementing the shallow atmosphere approximation in three dimensional finite element discretisations for dynamical cores. The approach makes use of the fact that the shallow atmosphere approximation metric can be…
Invariant discretization schemes are derived for the one- and two-dimensional shallow-water equations with periodic boundary conditions. While originally designed for constructing invariant finite difference schemes, we extend the usage of…
We define an abstract framework called {\it discrete finite differences embedding} which can be used to obtain discrete analogue of formal functional relations in the spirit of category theory. For ordinary differential equations we exhibit…
Ab initio electronic structure calculations of two-dimensional layered structures are typically performed using codes that were developed for three-dimensional structures, which are periodic in all three directions. The introduction of a…
In electronic structure theory, variational methods offer a valuable paradigm for approximating electronic ground states. However, for historical reasons, this principle is mostly restricted to model chemistries in pre-defined fixed basis…
Differential complexes such as the de Rham complex have recently come to play an important role in the design and analysis of numerical methods for partial differential equations. The design of stable discretizations of systems of partial…
We investigate the performance of algebraic multigrid methods for the solution of the linear system of equations arising from a Virtual Element discretization. We provide numerical experiments on very general polygonal meshes for a model…
I introduce an innovative methodology for deriving numerical models of systems of partial differential equations which exhibit the evolution of spatial patterns. The new approach directly produces a discretisation for the evolution of the…
This paper discusses the practical development of space-time boundary element methods for the wave equation in three spatial dimensions. The employed trial spaces stem from simplex meshes of the lateral boundary of the space-time cylinder.…
The aim of this paper is twofold. On the one hand, we test numerically the performance of mixed virtual elements in three dimensions for the first time in the literature to solve the mixed formulation of three-dimensional elliptic equations…
We review our recently developed electronic structure calculation methods used for the dynamics of large-scale solids or liquids with an efficient algorithm for large scale simultaneous linear equations. The electronic structure calculation…
We introduce a generalized finite difference method for solving a large range of fully nonlinear elliptic partial differential equations in three dimensions. Methods are based on Cartesian grids, augmented by additional points carefully…
This article aims to develop a direct numerical approach to solve the space-fractional partial differential equations (PDEs) based on a new differential quadrature (DQ) technique. The fractional derivatives are approximated by the weighted…
This article investigates a space-time differential model related to the degradation of stone artifacts caused by exposure to air and atmospheric agents, which specifically lead to the accumulation of salt crystals in the material. A…
This work develops and analyzes a variational-monolithic unfitted finite element formulation of a linear fluid-structure interaction problem in Eulerian coordinates with a fixed interface. The overall discretization is based on a backward…
We introduce a new Partition of Unity Method for the numerical homogenization of elliptic partial differential equations with arbitrarily rough coefficients. We do not restrict to a particular ansatz space or the existence of a finite…
The time discretization of stochastic spectral fractional wave equation is studied by using the difference methods. Firstly, we exploit rectangle formula to get a low order time discretization, whose the strong convergence order is smaller…
In this paper, we study a numerical method for the solution of partial differential equations on evolving surfaces. The numerical method is built on the stabilized trace finite element method (TraceFEM) for the spatial discretization and…
Finite difference method was extended to unstructured meshes to solve Euler equations. The spatial discretization is made of two steps. First, numerical fluxes are computed at the middle point of each edge with high order accuracy. In this…