Related papers: Smoothed Boundary Method for Solving Partial Diffe…
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…
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…
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…
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…
This article provides a general iterative approximation to partial differential equations, and thus establish existence of smooth solution. The heart of the method is to contract (or expand) the boundary conditions uniformly in the domain,…
We present a spectral method for parabolic partial differential equations with zero Dirichlet boundary conditions. The region {\Omega} for the problem is assumed to be simply-connected and bounded, and its boundary is assumed to be a smooth…
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…
Many physical systems of interest involve the close interaction of a flow in a domain with complex, time-varying boundaries. Treatment of boundaries of this nature is cumbersome due to the difficulty in explicitly tracking boundaries that…
A finite difference method is constructed to solve singularly perturbed convection-diffusion problems posed on smooth domains. Constraints are imposed on the data so that only regular exponential boundary layers appear in the solution. A…
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…
Analyzing complex fluid flow problems that involve multiple coupled domains, each with their respective set of governing equations, is not a trivial undertaking. Even more complicated is the elaborate and tedious task of specifying the…
In this paper, we analyze nonlinear differential equations subject to generalized boundary conditions. More specifically, we provide a framework from which we can provide conditions, which are straightforward to check, for the solvability…
In many applications, transport of particles can be described by the diffusion equation, or its convective-diffusion generalizations, in part of three-dimensional space. In particular, in surface deposition or in growth of aggregates or…
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…
Mixed boundary conditions are introduced to finite element exterior calculus. We construct smoothed projections from Sobolev de Rham complexes onto finite element de Rham complexes which commute with the exterior derivative, preserve…
In this paper we consider scalar parabolic equations in a general non-smooth setting with emphasis on mixed interface and boundary conditions. In particular, we allow for dynamics and diffusion on a Lipschitz interface and on the boundary,…
We prove nonlinear lower bounds and commutator estimates for the Dirichlet fractional Laplacian in bounded domains. The applications include bounds for linear drift-diffusion equations with nonlocal dissipation and global existence of weak…
A formulation of the boundary integral method for solving partial differential equations has been developed whereby the usual weakly singular integral and the Cauchy principal value integral can be removed analytically. The broad…
Stochastic diffusion equations are crucial for modeling a range of physical phenomena influenced by uncertainties. We introduce the generalized finite difference method for solving these equations. Then, we examine its consistency,…
Given only a collection of points sampled from a Riemannian manifold embedded in a Euclidean space, in this paper we propose a new method to solve elliptic partial differential equations (PDEs) supplemented with boundary conditions. Notice…