Related papers: Analysis of the diffuse-domain method for solving …
Extreme learning machine (ELM) is a methodology for solving partial differential equations (PDEs) using a single hidden layer feed-forward neural network. It presets the weight/bias coefficients in the hidden layer with random values, which…
The recent development of spectral method has been praised for its high-order convergence in simulating complex physical problems. The combination of embedded boundary method and spectral method becomes a mainstream way to tackle…
This paper enhances the Diffuse Interface Method (DIM) for simulating compressible multiphase flows across all Mach numbers by addressing the accuracy challenges posed at low Mach regimes. A correction to the Riemann solver is introduced,…
We adapt the Gradient Discretisation Method (GDM), originally designed for elliptic and parabolic partial differential equations, to the case of a linear scalar hyperbolic equations. This enables the simultaneous design and convergence…
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…
A new class of non-monotone finite difference (FD) approximation methods for approximating solutions to non-degenerate stationary Hamilton-Jacobi problems with Dirichlet boundary conditions is proposed and analyzed. The new FD methods add a…
A system of Boundary-Domain Integral Equations is derived from the mixed (Dirichlet-Neumann) boundary value problem for the diffusion equation in inhomogeneous media defined on an unbounded domain. Boundary-domain integral equations are…
Partial differential equations (PDEs) are widely used for modeling various physical phenomena. These equations often depend on certain parameters, necessitating either the identification of optimal parameters or the solution of the…
We derive novel algorithms for optimization problems constrained by partial differential equations describing multiscale particle dynamics, including non-local integral terms representing interactions between particles. In particular, we…
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…
Upcoming technologies like digital twins, autonomous, and artificial intelligent systems involving safety-critical applications require models which are accurate, interpretable, computationally efficient, and generalizable. Unfortunately,…
Meshing of geometric domains having curved boundaries by affine simplices produces a polytopial approximation of those domains. The resulting error in the representation of the domain limits the accuracy of finite element methods based on…
In this paper we present PDE and finite element analyses for a system of partial differential equations (PDEs) consisting of the Darcy equation and the Cahn-Hilliard equation, which arises as a diffuse interface model for the two phase…
This paper presents a theoretical discussion as well as novel solution algorithms for problems of scattering on smooth two-dimensional domains under Zaremba boundary conditions for which Dirichlet and Neumann conditions are specified on…
This work is concerned with the quantification of the epistemic uncertainties induced the discretization of partial differential equations. Following the paradigm of probabilistic numerics, we quantify this uncertainty probabilistically.…
We extend the Balancing Domain Decomposition by Constraints (BDDC) method to flows in porous media discretised by mixed-hybrid finite elements with combined mesh dimensions. Such discretisations appear when major geological fractures are…
In certain practical engineering applications, there is an urgent need to perform repetitive solving of partial differential equations (PDEs) in a short period. This paper primarily considers three scenarios requiring extensive repetitive…
Extreme learning machines (ELMs), which preset hidden layer parameters and solve for last layer coefficients via a least squares method, can typically solve partial differential equations faster and more accurately than Physics Informed…
We design a cut finite element method for the incompressible Stokes equations on curved domains. The cut finite element method allows for the domain boundary to cut through the elements of the computational mesh in a very general fashion.…
This paper studies the shallow Ritz method for solving one-dimensional diffusion-reaction problems. The method is capable of improving the order of approximation for non-smooth problems. By following a similar approach to the one presented…