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If a variational problem comes with no boundary conditions prescribed beforehand, and yet these arise as a consequence of the variation process itself, we speak of a free boundary values variational problem. Such is, for instance, the…

Differential Geometry · Mathematics 2017-03-14 Giovanni Moreno , Monika Ewa Stypa

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…

Numerical Analysis · Mathematics 2017-12-08 Brendan Keith , Socratis Petrides , Federico Fuentes , Leszek Demkowicz

We propose a new fictitious domain finite element method, well suited for elliptic problems posed in a domain given by a level-set function without requiring a mesh fitting the boundary. To impose the Dirichlet boundary conditions, we…

Numerical Analysis · Mathematics 2019-07-09 Michel Duprez , Alexei Lozinski

We present a new composite mesh finite element method for fluid--structure interaction problems. The method is based on surrounding the structure by a boundary-fitted fluid mesh which is embedded into a fixed background fluid mesh. The…

Numerical Analysis · Mathematics 2016-01-20 Andre Massing , Mats G. Larson , Anders Logg , Marie E. Rognes

We propose and analyze a general framework for space-time finite element methods that is based on least-squares finite element methods for solving a first-order reformulation of the thick parabolic obstacle problem. Discretizations based on…

Numerical Analysis · Mathematics 2025-03-12 José Joaquín Carvajal , Davood Damircheli , Thomas Führer , Francisco Fuica , Michael Karkulik

We consider locally stabilized, conforming finite element schemes on completely unstructured simplicial space-time meshes for the numerical solution of parabolic initial-boundary value problems with variable, possibly discontinuous in space…

Numerical Analysis · Mathematics 2020-03-23 Ulrich Langer , Andreas Schafelner

In this paper we propose a new method to stabilise non-symmetric indefinite problems. The idea is to solve a forward and an adjoint problem simultaneously using a suitable stabilised finite element method. Both stabilisation of the element…

Numerical Analysis · Mathematics 2013-08-05 Erik Burman

The finite element simulation of dynamic wetting phenomena, requiring the computation of flow in a domain confined by intersecting a liquid-fluid free surface and a liquid-solid interface, with the three-phase contact line moving across the…

Computational Physics · Physics 2012-02-20 J. E. Sprittles , Y. D. Shikhmurzaev

We consider an interface problem often arising in transport problems: a coupled system of partial differential equations with one (elliptic) transport equation on a bounded domain and one equation (in this case the Laplace problem) on the…

Numerical Analysis · Mathematics 2016-05-24 Christoph Erath , Robert Schorr

An isogeometric boundary element method for problems in elasticity is presented, which is based on an independent approximation for the geometry, traction and displacement field. This enables a flexible choice of refinement strategies,…

Numerical Analysis · Computer Science 2015-02-04 Benjamin Marussig , Jürgen Zechner , Gernot Beer , Thomas-Peter Fries

A finite element approach for approximating the solution of a mathematical model for the response of a penetrable, bounded object (obstacle) to the excitation by an external electromagnetic field is presented and investigated. The model…

Numerical Analysis · Mathematics 2026-04-16 Lutz Angermann

The finite element method can be viewed as a machine that automates the discretization of differential equations, taking as input a variational problem, a finite element and a mesh, and producing as output a system of discrete equations.…

Numerical Analysis · Mathematics 2011-12-05 Anders Logg

We consider within a finite element approach the usage of different adaptively refined meshes for different variables in systems of nonlinear, time-depended PDEs. To resolve different solution behaviours of these variables, the meshes can…

Numerical Analysis · Mathematics 2010-05-27 Thomas Witkowski , Axel Voigt

We present a cut finite element method for the heat equation on two overlapping meshes: a stationary background mesh and an overlapping mesh that moves around inside/"on top" of it. Here the overlapping mesh is prescribed a simple…

Numerical Analysis · Mathematics 2023-03-02 Mats G. Larson , Anders Logg , Carl Lundholm

In this paper, we present a new immersed finite element scheme for solving elliptic interface problems on unfitted meshes by combining the skeletal finite element method (FEM) with the standard FEM. The skeletal FEM is used for the…

Numerical Analysis · Mathematics 2025-09-17 Lin Yang , Qilong Zhai

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…

Numerical Analysis · Mathematics 2017-10-11 Peter Hansbo , Mats G. Larson , Andre Massing

This work develops and analyzes a variational-monolithic unfitted finite element formulation of a linear fluid-structure interaction problem in Eulerian coordinates with a fixed interface. The overall discretization is based on a backward…

Numerical Analysis · Mathematics 2026-03-27 Stefan Frei , Tobias Knoke , Marc C. Steinbach , Anne-Kathrin Wenske , Thomas Wick

In this contribution we present a new computational method for coupled bulk-surface problems on time-dependent domains. The method is based on a space-time formulation using discontinuous piecewise linear elements in time and continuous…

Numerical Analysis · Mathematics 2016-05-25 Peter Hansbo , Mats G. Larson , Sara Zahedi

Compressible Mooney-Rivlin theory has been used to model hyperelastic solids, such as rubber and porous polymers, and more recently for the modeling of soft tissues for biomedical tissues, undergoing large elastic deformations. We propose a…

Numerical Analysis · Mathematics 2025-10-20 Suzanne M. Shontz , Stephen A. Vavasis

In this paper, the interaction between two immiscible fluids with a finite mobility ratio is investigated numerically within a Hele-Shaw cell. Fingering instabilities initiated at the interface between a low viscosity fluid and a high…

Fluid Dynamics · Physics 2021-04-01 S. J. Jackson , D. Stevens , H. Power , D. Giddings