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We present and analyze two stabilized finite element methods for solving numerically the Poisson--Nernst--Planck equations. The stabilization we consider is carried out by using a shock detector and a discrete graph Laplacian operator for…
The integral equation approach to partial differential equations (PDEs) provides significant advantages in the numerical solution of the incompressible Navier-Stokes equations. In particular, the divergence-free condition and boundary…
This work investigates two physics-based models that simulate the non-linear partial differential algebraic equations describing an electric double layer supercapacitor. In one model the linear dependence between electrolyte concentration…
Faceted shapes, such as polyhedra, are commonly found in systems of nanoscale, colloidal, and granular particles. Many interesting physical phenomena, like crystal nucleation and growth, vacancy motion, and glassy dynamics are challenging…
We consider the finite element method on locally damaged meshes allowing for some distorted cells which are isolated from one another. In the case of the Poisson equation and piecewise linear Lagrange finite elements, we show that the usual…
We develop a theoretical framework for the analysis of stabilized cut finite element methods for the Laplace-Beltrami operator on a manifold embedded in $\mathbb{R}^d$ of arbitrary codimension. The method is based on using continuous…
The paper introduces a finite element method for the incompressible Navier--Stokes equations posed on a closed surface $\Gamma\subset\R^3$. The method needs a shape regular tetrahedra mesh in $\mathbb{R}^3$ to discretize equations on the…
We propose a new unfitted finite element method for simulation of two-phase flows in presence of insoluble surfactant. The key features of the method are 1) discrete conservation of surfactant mass; 2) the possibility of having meshes that…
We propose algorithms for solving convective-diffusion partial differential equations (PDEs), which model surfactant concentration and heat transport on evolving surfaces, based on intrinsic kernel-based meshless collocation methods. The…
The multiscale complexity of modern problems in computational science and engineering can prohibit the use of traditional numerical methods in multi-dimensional simulations. Therefore, novel algorithms are required in these situations to…
Stochastic partial differential equations (SPDEs) represent a very active research field with numerous recent developments and breakthrough results. There are several well-established approaches and methods used to construct solutions for…
This article is focused on two related topics within the study of partial differential equations (PDEs) that illustrate a beautiful connection between dynamics, topology, and analysis: stability and spatial dynamics. The first is a property…
In this paper we consider a fully discrete numerical method for the unsteady Navier-Stokes equations on a smooth closed stationary surface in $\mathbb{R}^3$. We use the surface finite element method (SFEM) with a generalized Taylor-Hood…
We develop a mesh-free, derivative-free, matrix-free, and highly parallel localized stochastic method for high-dimensional semilinear parabolic PDEs. The efficiency of the proposed method is built upon four essential components: (i) a…
Recently, the $P_1$-nonconforming finite element space over square meshes has been proved stable to solve Stokes equations with the piecewise constant space for velocity and pressure, respectively. In this paper, we will introduce its…
The Finite Element Method (FEM) is a powerful computational tool for solving partial differential equations (PDEs). Although commercial and open-source FEM software packages are widely available, an independent implementation of FEM…
Surface partial differential equations arise in numerous scientific and engineering applications. Their numerical solution on static and evolving surfaces remains challenging due to geometric complexity and, for evolving geometries, the…
The paper introduces a new finite element numerical method for the solution of partial differential equations on evolving domains. The approach uses a completely Eulerian description of the domain motion. The physical domain is embedded in…
Spectral methods are an important part of scientific computing's arsenal for solving partial differential equations (PDEs). However, their applicability and effectiveness depend crucially on the choice of basis functions used to expand the…
Piecewise divergence-free nonconforming virtual elements are designed for Stokes problem in any dimensions. After introducing a local energy projector based on the Stokes problem and the stabilization, a divergence-free nonconforming…