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This article initiates the study of space-time adaptive mesh refinements for time-dependent boundary element formulations of wave equations. Based on error indicators of residual type, we formulate an adaptive boundary element procedure for…

We suggest a novel shape matching algorithm for three-dimensional surface meshes of disk or sphere topology. The method is based on the physical theory of nonlinear elasticity and can hence handle large rotations and deformations.…

In this paper, we define new unfitted finite element methods for numerically approximating the solution of surface partial differential equations using bulk finite elements. The key idea is that the $n$-dimensional hypersurface, $\Gamma…

Nonlinear two-point boundary value problems arise in numerous areas of application. The existence and number of solutions for various cases has been studied from a theoretical standpoint. These results generally rely upon growth conditions…

This work develops a dynamic homogenization approach for metamaterials. It finds an approximate macroscopic homogenized equation with constant coefficients posed in space and time; however, the resulting homogenized equation is higher order…

Nonlocal models provide exceptional simulation fidelity for a broad spectrum of scientific and engineering applications. However, wider deployment of nonlocal models is hindered by several modeling and numerical challenges. Among those, we…

This work describes a Bayesian framework for reconstructing the boundaries that represent targeted features in an image, as well as the regularity (i.e., roughness vs. smoothness) of these boundaries.This regularity often carries crucial…

Combined-resolution simulations are an effective way to study molecular properties across a range of length- and time-scales. These simulations can benefit from adaptive boundaries that allow the high-resolution region to adapt (change size…

Amorphous materials of homogeneous structures usually suffer from nonuniform deformation under shear, which can develop into shear localization and eventually destructive shear band. One approach to tackle this issue is to introduce an…

We present a strongly-coupled immersed-boundary method for flow-structure interaction problems involving thin deforming bodies. The method is stable for arbitrary choices of solid-to-fluid mass ratios and for large body motions. As with…

We present a method of CutFEM type for the Poisson problem with either Dirichlet or Neumann boundary conditions. The computational mesh is obtained from a background (typically uniform Cartesian) mesh by retaining only the elements…

The analysis and homogenization of a moving boundary problem for a highly heterogeneous, periodic two-phase medium is considered. In this context, the normal velocity governing the motion of the interface separating the two competing phases…

Introducing flexibility in the time-discretisation mesh can improve convergence and computational time when solving differential equations numerically, particularly when the solutions are discontinuous, as commonly found in control problems…

We propose a patchwise local Fourier extension method for approximating smooth functions on general two dimensional domains with curved boundaries. The domain is embedded into a Cartesian background grid and decomposed into rectangular…

Finite elasticity problems commonly include material and geometric nonlinearities and are solved using various numerical methods. However, for highly nonlinear problems, achieving convergence is relatively difficult and requires small load…

Finite element methods provide accurate and efficient methods for the numerical solution of partial differential equations by means of restricting variational problems to finite-dimensional approximating spaces. However, they do not…

Chaotic free surface flows are challenging problems to simulate numerically, mainly due to the significant changes in geometry and frequent topological changes. Methods that track the evolution of the fluid in a Lagrangian formulation are a…

Localized features such as singularities, sharp gradients, discontinuities, and moving sources require adaptive finite element discretizations. Conventional refinement strategies introduce significant computational overhead through…

This paper presents an asymptotically compatible error bound for the finite element method (FEM) applied to a nonlocal diffusion model. The analysis covers two scenarios: meshes with and without shape regularity. For shape-regular meshes,…

This work presents a new approach to efficiently model the cathode in the moving boundary value problem of electrochemical machining. Until recently, the process simulation with finite elements had the drawback of remeshing required by the…