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We introduce a diffuse interface box method (DIBM) for the numerical approximation on complex geometries of elliptic problems with Dirichlet boundary conditions. We derive a priori $H^1$ and $L^2$ error estimates highlighting the r\^{o}le…
Here we study the Dirichlet problem for first order linear and quasi-linear hyperbolic PDEs on a simply connected bounded domain of $\R^2$, where the domain has an interior outflow set and a mere inflow boundary. By means of a Lyapunov…
In this paper, we discuss the steady and time-dependent nonlinear convection-diffusion (advection-diffusion) equations with the Dirichlet boundary condition. For the steady nonlinear equation, we use an iteration method to reformulate the…
In this paper, we propose simple numerical algorithms for partial differential equations (PDEs) defined on closed, smooth surfaces (or curves). In particular, we consider PDEs that originate from variational principles defined on the…
In this paper, we propose a numerical method to approximate the solution of partial differential equations in irregular domains with no-flux boundary conditions by means of spectral methods. The main features of this method are its…
In this paper we study the Poisson and heat equations on bounded and unbounded domains with smooth boundary with random Dirichlet boundary conditions. The main novelty of this work is a convenient framework for the analysis of such…
Many geometry processing techniques require the solution of partial differential equations (PDEs) on manifolds embedded in $\mathbb{R}^2$ or $\mathbb{R}^3$, such as curves or surfaces. Such manifold PDEs often involve boundary conditions…
In this article, we propose novel boundary treatment algorithms to avoid order reduction when implicit-explicit Runge-Kutta time discretization is used for solving convection-diffusion-reaction problems with time-dependent Di\-richlet…
Shape calculus concerns the calculation of directional derivatives of some quantity of interest, typically expressed as an integral. This article introduces a type of shape calculus based on localized dilation of boundary faces through…
In theory, diffusion curves promise complex color gradations for infinite-resolution vector graphics. In practice, existing realizations suffer from poor scaling, discretization artifacts, or insufficient support for rich boundary…
Anomalous diffusion is a phenomenon that cannot be modeled accurately by second-order diffusion equations, but is better described by fractional diffusion models. The nonlocal nature of the fractional diffusion operators makes substantially…
We consider fully discrete embedded finite element approximations for a shallow water hyperbolic problem and its reduced-order model. Our approach is based on a fixed background mesh and an embedded reduced basis. The Shifted Boundary…
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…
Neural operator learning accelerates PDE solution by approximating operators as mappings between continuous function spaces. Yet in many engineering settings, varying geometry induces discrete structural changes, including topological…
Modeling and simulation of multiphase flows in complex geomerties are challenging due to the complexity in describing the interface topology changes among different phases and the difficulty in implementing the boundary conditions on the…
Problems with localized nonhomogeneous material properties present well-known challenges for numerical simulations. In particular, such problems may feature large differences in length scales, causing difficulties with meshing and…
Propagation characteristics of a wave are defined by the dispersion relationship, from which the governing partial differential equation (PDE) can be recovered. PDEs are commonly solved numerically using the finite-difference (FD) method,…
The objective of this study is to address the difficulty of simplifying the geometric model in which a differential problem is formulated, also called defeaturing, while simultaneously ensuring that the accuracy of the solution is…
In this paper, we develop a family of high order asymptotic preserving schemes for some discrete-velocity kinetic equations under a diffusive scaling, that in the asymptotic limit lead to macroscopic models such as the heat equation, the…
We present a numerical approximation method for linear diffusion-reaction problems with possibly discontinuous Dirichlet boundary conditions. The solution of such problems can be represented as a linear combination of explicitly known…