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We describe an exact and highly efficient numerical algorithm for solving a special but important class of convection-diffusion equations. These equations occur in many problems in physics, chemistry, or biology, and they are usually hard…
In this article we study the numerical approximation of a variable coefficient fractional diffusion equation. Using a change of variable, the variable coefficient fractional diffusion equation is transformed into a constant coefficient…
We consider a filter stabilization technique with a deconvolution-based indicator function for the simulation of advection dominated advection-diffusion-reaction (ADR) problems with under-refined meshes. The proposed technique has been…
In this work, we discuss and compare three methods for the numerical approximation of constant- and variable-coefficient diffusion equations in both single and composite domains with possible discontinuity in the solution/flux at…
In this work, we derive particle schemes, based on micro-macro decomposition, for linear kinetic equations in the diffusion limit. Due to the particle approximation of the micro part, a splitting between the transport and the collision part…
Heterogeneous anisotropic diffusion problems arise in the various areas of science and engineering including plasma physics, petroleum engineering, and image processing. Standard numerical methods can produce spurious oscillations when they…
In this paper, we propose and analyze a new stochastic homogenization method for diffusion equations with random and fast oscillatory coefficients. In the proposed method, the homogenized solutions are sought through a two-stage procedure.…
In this paper, a semi-discrete spatial finite volume (FV) method is proposed and analyzed for approximating solutions of anomalous subdiffusion equations involving a temporal fractional derivative of order $\alpha \in (0,1)$ in a…
This paper presents a new method to approximate the time-dependent convection-diffusion equations using conforming finite element methods, ensuring that the discrete solution respects the physical bounds imposed by the differential…
We propose and analyze a new method for the unsteady incompressible magnetohydrodynamics equations on convex domains with hybrid approximations of both vector-valued and scalar-valued fields. The proposed method is convection-semirobust,…
A finite element method of any order is applied on a Bakhvalov-type mesh to solve a singularly perturbed convection--diffusion equation in 2D, whose solution exhibits exponential boundary layers. A uniform convergence of (almost) optimal…
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…
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…
Fractional equations have become the model of choice in several applications where heterogeneities at the microstructure result in anomalous diffusive behavior at the macroscale. In this work we introduce a new fractional operator…
We introduce a family of hybrid discretisations for the numerical approximation of optimal control problems governed by the equations of immiscible displacement in porous media. The proposed schemes are based on mixed and discontinuous…
We develop finite element methods for coupling the steady-state Onsager--Stefan--Maxwell equations to compressible Stokes flow. These equations describe multicomponent flow at low Reynolds number, where a mixture of different chemical…
In this paper, we present a model based on a local thermodynamic equilibrium, weakly ionized plasma-mixture model used for medical and technical applications in etching processes. We consider a simplified model based on the Maxwell-Stefan…
In this paper, we derive two bound-preserving and mass-conserving schemes based on the fractional-step method and high-order compact (HOC) finite difference method for nonlinear convection-dominated diffusion equations. We split the…
We introduce new adaptive artificial anti-diffusion (AAAD) methods for one- and two-dimensional hyperbolic systems of conservation laws. The key idea is to reduce the amount of numerical dissipation present in a given numerical method by…
This paper introduces a novel approach to approximate a broad range of reaction-convection-diffusion equations using conforming finite element methods while providing a discrete solution respecting the physical bounds given by the…