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We introduce discontinuous spectral-element methods of arbitrary order that are well balanced, conservative of mass, and conservative or dissipative of total energy (i.e., a mathematical entropy function) for a covariant flux formulation of…
In this paper we analyze a space-time unfitted finite element method for the discretization of scalar surface partial differential equations on evolving surfaces. For higher order approximations of the evolving surface we use the technique…
Spectral methods for solving partial differential equations (PDEs) and stochastic partial differential equations (SPDEs) often use Fourier or polynomial spectral expansions on either uniform and non-uniform grids. However, while very widely…
A family of Virtual Element Methods for the 2D Navier-Stokes equations is proposed and analysed. The schemes provide a discrete velocity field which is point-wise divergence-free. A rigorous error analysis is developed, showing that the…
We propose an efficient and accurate parametric finite element method (PFEM) for solving sharp-interface continuum models for solid-state dewetting of thin films with anisotropic surface energies. The governing equations of the…
In this paper, we propose a mesh-free numerical method for solving elliptic PDEs on unknown manifolds, identified with randomly sampled point cloud data. The PDE solver is formulated as a spectral method where the test function space is the…
Solving partial differential equations (PDEs) on shapes underpins many shape analysis and engineering tasks; yet, prevailing PDE solvers operate on polygonal/triangle meshes while modern 3D assets increasingly live as neural…
A new method for solving numerically stochastic partial differential equations (SPDEs) with multiple scales is presented. The method combines a spectral method with the heterogeneous multiscale method (HMM) presented in [W. E, D. Liu, and…
The intrusive (sample-free) spectral stochastic finite element method (SSFEM) is a powerful numerical tool for solving stochastic partial differential equations (PDEs). However, it is not widely adopted in academic and industrial…
Machine learning based partial differential equations (PDEs) solvers have received great attention in recent years. Most progress in this area has been driven by deep neural networks such as physics-informed neural networks (PINNs) and…
The numerical solution of stochastic partial differential equations (SPDE) presents challenges not encountered in the simulation of PDEs or SDEs. Indeed, the roughness of the noise in conjunction with nonlinearities in the drift typically…
The subject of this paper is the design of efficient and stable spectral methods for time-dependent partial differential equations in unit balls. We commence by sketching the desired features of a spectral method, which is defined by a…
In this paper we consider the numerical approximation of the incompressible surface Navier--Stokes equations on an evolving surface. For the discrete representation of the moving surface we use parametric finite elements of degree $\ell…
We consider the Navier-Stokes equations in a channel with a narrowing of varying height. The model is discretized with high-order spectral element ansatz functions, resulting in 6372 degrees of freedom. The steady-state snapshot solutions…
This article presents a higher-order spectral element method for the two-dimensional Stokes interface problem involving a piecewise constant viscosity coefficient. The proposed numerical formulation is based on least-squares formulation.…
Spectral element methods (SEM), which are extensions of finite element methods (FEM), are important emerging techniques for solving partial differential equations in physics and engineering. SEM can potentially deliver better accuracy due…
Efficient and stable solution of partial differential equations (PDEs) is central to scientific and engineering applications, yet existing numerical solvers rely heavily on matrix based discretizations, while learning based methods require…
Deep models have achieved impressive progress in solving partial differential equations (PDEs). A burgeoning paradigm is learning neural operators to approximate the input-output mappings of PDEs. While previous deep models have explored…
An important step in shape optimization with partial differential equation constraints is to adapt the geometry during each optimization iteration. Common strategies are to employ mesh-deformation or re-meshing, where one or the other…
We develop a cut finite element method for the Darcy problem on surfaces. The cut finite element method is based on embedding the surface in a three dimensional finite element mesh and using finite element spaces defined on the three…