An unfitted finite element method, based on Nitsche's method, for elliptic interface problems

By Anita Hansbo, Peter Hansbo. 2002

964 citations recorded by the source corpus.

In this paper we propose a method for the finite element solution of elliptic interface problem, using an approach due to Nitsche. The method allows for discontinuities, internal to the elements, in the approximation across the interface. We show that optimal order of convergence holds without restrictions on the location of the interface relative to the mesh. Further, we derive a posteriori error estimates for the purpose of controlling functionals of the error and present some numerical examples.

Subfields: Computational Mechanics, Computational Theory and Mathematics, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/1f37534716add06d7b40480460062a1daf91c457

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In Computational Mechanics

In Computational Theory and Mathematics

In Mechanics of Materials

Citations in the corpus

Stable imposition of stiff constraints in explicit dynamics for embedded finite element methods (2012, background): “This approach was pursued by researchers working on the fictitious domain methods (see Glowinski et al. [10], Burman and Hansbo [11]) as well as the finite difference immersed interface methods of Bedrossian et al. [12].”
On strategies for enforcing interfacial constraints and evaluating jump conditions with the extended finite element method (2004, methodology): “Many investigations into elliptic interface problems take this field as given (see, for example, Reference [18], or Reference [26]).”
An extended finite element method with analytical enrichment for cohesive crack modeling (2009, background): “Hansbo and Hansbo [40] presented a different formulation for representing strong discontinuities in elliptic interface problems, demonstrating it with two-dimensional stationary heat condition problems, and later [41] extended the work to elasticity providing an a priori error analysis, and examples”
The Finite Cell Method: A Review in the Context of Higher-Order Structural Analysis of CAD and Image-Based Geometric Models (2015, methodology): “Possible technologies are the penalty method [9,55,199], the Lagrange multiplier method [29,55,56,58,59,61,201] or Nitsche’s method [17,30,43,51,53,62,64,66].”
Analysis of trace finite element methods for surface partial differential equations (2015, background): “A (slightly) simplified version of the following lemma is presented in Hansbo & Hansbo (2002, 2004).”
Robust imposition of Dirichlet boundary conditions on embedded surfaces (2012, methodology): “More importantly, from a stability perspective for the method, such a choice of the stabilization parameter, which ensures the positive definiteness of the bilinear form, also facilitates mathematical analysis and establishes proofs of optimal-order convergence [9, 19, 26].”
A Nitsche extended finite element method for incompressible elasticity with discontinuous modulus of elasticity (2009, methodology): “We need the following inverse inequality, for a proof of which we refer to [5].”
Robust flux error estimation of an unfitted Nitsche method for high-contrast interface problems (2016, neutral): “See for example [2], [22], [8], [11], [16], [14], [4], [9], [5], [18], [17], [23], [7], [3], [21], [1], [19], [15].”
Discontinuous Galerkin methods for the Stokes equations using divergence‐free approximations (2008, background): “In fact, it is worth noting the contributions in solid mechanics by Hansbo and co-workers [14, 15 , 12], which have inspired several authors (see for instance [16] for the solution of the Navier-Stokes equations) and in particular this paper.”
An Isogeometric design-through-analysis methodology based on adaptive hierarchical refinement of NURBS, immersed boundary methods, and T-spline CAD surfaces (2012, methodology): “At all boundaries, we impose Ψ strongly, while its derivatives are imposed weakly by a variant of Nitsche’s method [79, 81] with a penalty parameter β = 50.”