Related papers: Optimal error estimates of the diffuse domain meth…
In this paper, we mainly discuss the convergence behavior of diffuse domain method (DDM) for solving semilinear parabolic equations with Neumann boundary condition defined in general irregular domains. We use a phasefield function to…
In recent work, Li et al.\ (Comm.\ Math.\ Sci., 7:81-107, 2009) developed a diffuse-domain method (DDM) for solving partial differential equations in complex, dynamic geometries with Dirichlet, Neumann, and Robin boundary conditions. The…
Diffuse domain methods (DDMs) have gained significant attention for solving partial differential equations (PDEs) on complex geometries. These methods approximate the domain by replacing sharp boundaries with a diffuse layer of thickness…
The diffuse domain method for partial differential equations on complicated geometries recently received strong attention in particular from practitioners, but many fundamental issues in the analysis are still widely open. In this paper we…
We use a diffuse interface method for solving Poisson's equation with a Dirichlet condition on an embedded curved interface. The resulting diffuse interface problem is identified as a standard Dirichlet problem on approximating regular…
Diffuse domain methods (DDMs) have garnered significant attention for approximating solutions to partial differential equations on complex geometries. These methods implicitly represent the geometry by replacing the sharp boundary interface…
The solution of partial differential equations (PDEs) on complex domains often presents a significant computational challenge by requiring the generation of fitted meshes. The Diffuse Domain Method (DDM) is an alternative which reformulates…
We present a waveform relaxation version of the Dirichlet-Neumann and Neumann-Neumann methods for parabolic problems. Like the Dirichlet-Neumann method for steady problems, the method is based on a non-overlapping spatial domain…
In this paper, we consider the finite element approximation for a parabolic problem on a smooth domain $\Omega \subset \mathbb{R}^N$ with the inhomogeneous Neumann boundary condition. We emphasize that the domain can be non-convex in…
The diffuse-domain, or smoothed boundary, method is an attractive approach for solving partial differential equations in complex geometries because of its simplicity and flexibility. In this method the complex geometry is embedded into a…
In this article, we have studied the convergence behavior of the Dirichlet-Neumann waveform relaxation algorithms for time-fractional sub-diffusion and diffusion wave equations in 1D \& 2D for regular domains, where the dimensionless…
In this paper we consider 2D nonlocal diffusion models with a finite nonlocal horizon parameter $\delta$ characterizing the range of nonlocal interactions, and consider the treatment of Neumann-like boundary conditions that have proven…
We study the behaviour of solutions of linear non-autonomous parabolic equations subject to Dirichlet or Neumann boundary conditions under perturbation of the domain. We prove that Mosco convergence of function spaces for non-autonomous…
In this article, we describe an approach for solving partial differential equations with general boundary conditions imposed on arbitrarily shaped boundaries. A continuous function, the domain parameter, is used to modify the original…
The gradient discretisation method (GDM) is a generic framework designed recently, as a discretise in spatial space, to partial differential equations. This paper aims to use the GDM to establish a first general error estimate for numerical…
We analyse a diffuse interface type approximation, known as the diffuse domain approach, of a linear coupled bulk-surface elliptic partial differential system. The well-posedness of the diffuse domain approximation is shown using weighted…
In this article, we have studied the convergence behavior of the Dirichlet-Neumann and Neumann- Neumann waveform relaxation algorithms for time-fractional sub-diffusion and diffusion-wave equations in 1D & 2D for regular domains, where the…
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…
The article discusses the gradient discretisation method (GDM) for distributed optimal control problems governed by diffusion equation with pure Neumann boundary condition. Using the GDM framework enables to develop an analysis that…
This paper explores the convergence behavior of two waveform relaxation algorithms, namely the Dirichlet-Neumann and Neumann-Neumann Waveform Relaxation algorithms, for an optimal control problem with a sub-diffusion partial differential…