Related papers: Imposing jump conditions on nonconforming interfac…
In this paper we develop finite difference schemes for elliptic problems with piecewise continuous coefficients that have (possibly huge) jumps across fixed internal interfaces. In contrast with such problems involving one smooth…
We study finite element approximations of elliptic and parabolic interface problems with discontinuous coefficients and nonlinear jump conditions. We introduce a scalar interface reduction in which the solution is decomposed into a…
A novel sharp interface ghost-cell based immersed boundary method has been proposed and its parameters have been optimized against an analytical model in diffusion applications. The proposed embedded constrained moving least-squares (ECMLS)…
Motivated by the need for efficient estimation of conditional expectations, we consider a least-squares function approximation problem with heavily polluted data. Existing methods that are effective in the small-noise regime are suboptimal…
Immersed boundary methods are extensively used for simulations of dynamic solid objects interacting with fluids due to their computational efficiency and modelling flexibility compared to body-fitted grid methods. However, thin geometries,…
We consider the discretization of a stationary Stokes interface problem in a velocity-pressure formulation. The interface is described implicitly as the zero level of a scalar function as it is common in level set based methods. Hence, the…
Coarse grid projection (CGP) is a multiresolution technique for accelerating numerical calculations associated with a set of nonlinear evolutionary equations along with the stiff Poisson equations. In this article we use CGP for the first…
The convergence and optimality of adaptive mixed finite element methods for the Poisson equation are established in this paper. The main difficulty for mixed finite element methods is the lack of minimization principle and thus the failure…
The Fast Multipole Method (FMM) provides a highly efficient computational tool for solving constant coefficient partial differential equations (e.g. the Poisson equation) on infinite domains. The solution to such an equation is given as the…
This paper presents and analyzes an immersed finite element (IFE) method for solving Stokes interface problems with a piecewise constant viscosity coefficient that has a jump across the interface. In the method, the triangulation does not…
Physics-informed neural networks (PINNs) have emerged as a flexible framework for solving partial differential equations, but their performance on interface problems remains challenging because continuity and flux conditions are typically…
Fast Fourier Transform (FFT)-based solvers for the Poisson equation are highly efficient, exhibiting $O(N\log N)$ computational complexity and excellent parallelism. However, their application is typically restricted to simple, regular…
In this paper, we present proofs of the coerciveness of first-order system least-squares methods for general (possibly indefinite) second-order linear elliptic PDEs under a minimal uniqueness assumption. For general linear second-order…
We present a new higher-order accurate finite difference explicit jump Immersed Interface Method (HEJIIM) for solving two-dimensional elliptic problems with singular source and discontinuous coefficients in the irregular region on a compact…
A finite element methodology for large classes of variational boundary value problems is defined which involves discretizing two linear operators: (1) the differential operator defining the spatial boundary value problem; and (2) a Riesz…
A novel unconstrained optimization model named weighted trace-penalty minimization (WTPM) is proposed to address the extreme eigenvalue problem arising from the Full Configuration Interaction (FCI) method. Theoretical analysis shows that…
One of the major issues in the computational mechanics is to take into account the geometrical complexity. To overcome this difficulty and to avoid the expensive mesh generation, geometrically unfitted methods, i.e. the numerical methods…
Flow matching (FM) is a family of training algorithms for fitting continuous normalizing flows (CNFs). Conditional flow matching (CFM) exploits the fact that the marginal vector field of a CNF can be learned by fitting least-squares…
We present a framework for solving partial different equations on evolving surfaces. Based on the grid-based particle method (GBPM) [18], the method can naturally resample the surface even under large deformation from the motion law. We…
A projection-based formulation is presented for non-linear model reduction of problems with extreme scale disparity. The approach allows for the selection of an arbitrary, but complete, set of solution variables while preserving the…