Related papers: A space-time FEM for PDEs on evolving surfaces
Spatial reaction-diffusion models have been employed to describe many emergent phenomena in biological systems. The modelling technique most commonly adopted in the literature implements systems of partial differential equations (PDEs),…
As fractional diffusion equations can describe the early breakthrough and the heavy-tail decay features observed in anomalous transport of contaminants in groundwater and porous soil, they have been commonly employed in the related…
In fractured natural formations, the equations governing fluid flow and geomechanics are strongly coupled. Hydrodynamical properties depend on the mechanical configuration, and they are therefore difficult to accurately resolve using…
We introduce a general framework for solving partial differential equations (PDEs) using generative diffusion models. In particular, we focus on the scenarios where we do not have the full knowledge of the scene necessary to apply classical…
In this paper, the surface of revolution discrete element method (SR-DEM) is introduced to simulate systems of particles with closed surfaces of revolution. Due to the cylindrical symmetry of a surface of revolution, the geometry of any…
The paper introduces a finite element method for the incompressible Navier--Stokes equations posed on a closed surface $\Gamma\subset\R^3$. The method needs a shape regular tetrahedra mesh in $\mathbb{R}^3$ to discretize equations on the…
We prove optimal error bounds for a second order in time finite element approximation of curve shortening flow in possibly higher codimension. In addition, we introduce a second order in time method for curve diffusion. Both schemes are…
This paper presents an extension of the discrete element method using a phase-field formulation to incorporate grain shape and its evolution. The introduction of a phase variable enables an effective representation of grain geometry and…
This year marks the eightieth anniversary of the invention of the finite element method (FEM). FEM has become the computational workhorse for engineering design analysis and scientific modeling of a wide range of physical processes,…
The aim of this paper is to develop and analyze numerical schemes for approximately solving the backward problem of subdiffusion equation involving a fractional derivative in time with order $\alpha\in(0,1)$. After using quasi-boundary…
We develop a family of expanded mixed Multiscale Finite Element Methods (MsFEMs) and their hybridizations for second-order elliptic equations. This formulation expands the standard mixed Multiscale Finite Element formulation in the sense…
A three-dimensional simulation model is proposed here to study the erosive wear of structure caused by solid particles, which accounts for the accumulation of surface deformation and degradation during the erosion process. Although there…
In this paper, we introduce the concept of Developmental Partial Differential Equation (DPDE), which consists of a Partial Differential Equation (PDE) on a time-varying manifold with complete coupling between the PDE and the manifold's…
We propose an adaptive finite element method to approximate the solutions to reaction-diffusion systems on time-dependent domains and surfaces. We derive a computable error estimator that provides an upper bound for the error in the…
This paper proposes the response surface method for finite element model updating. The response surface method is implemented by approximating the finite element model surface response equation by a multi-layer perceptron. The updated…
In this paper, we develop a high order numerical method for the numerical solutions of scattering problems with slightly perturbed periodic surfaces in two dimensional spaces. Based on the regularity property introduced in Part I, the…
This paper presents a novel methodology for fast simulation and analysis of transient heat transfer. The proposed methodology is suitable for real-time applications owing to (i) establishing the solution method from the viewpoint of…
We develop a domain-decomposition model reduction method for linear steady-state convection-diffusion equations with random coefficients. Of particular interest to this effort are the diffusion equations with random diffusivities, and the…
We present a method to simulate fluid flow on evolving surfaces, e.g., an oil film on a water surface. Given an animated surface (e.g., extracted from a particle-based fluid simulation) in three-dimensional space, we add a second simulation…
We present and analyze a semi-discrete finite element scheme for a system consisting of a geometric evolution equation for a curve and a parabolic equation on the evolving curve. More precisely, curve shortening flow with a forcing term…