A finite element method for interface problems in domains with smooth boundaries and interfaces

Bramble, James H.; King, J. Thomas

doi:10.1007/bf02127700

Cited by 273 publications

(169 citation statements)

References 14 publications

Supporting

Mentioning

168

Contrasting

Order By: Relevance

“…For second order elliptic interface problems, it is well known that standard finite element approximations can converge slowly [18]. Optimal convergence rates can be restored if the finite element mesh is aligned to the interface [19][20][21]. For complicated interface geometries the construction of such a mesh may be very difficult; therefore, many researchers have developed numerical methods to treat interface problems.…”

Section: Resultsmentioning

confidence: 99%

Computational Issues in Sensitivity Analysis for 1D Interface Problems

Davis

Singler

2011

JCM

View full text Add to dashboard Cite

This paper is concerned with the construction of accurate and efficient computational algorithms for the numerical approximation of sensitivities with respect to a parameter dependent interface location. Motivated by sensitivity analysis with respect to piezoelectric actuator placement on an Euler-Bernoulli beam, this work illustrates the key concepts related to sensitivity equation formulation for interface problems where the parameter of interest determines the location of the interface. A fourth order model problem is considered, and a homogenization procedure for sensitivity computation is constructed using standard finite element methods. Numerical results show that proper formulation and approximation of the sensitivity interface conditions is critical to obtaining convergent numerical sensitivity approximations. A second order elliptic interface model problem is also mentioned, and the homogenization procedure is outlined briefly for this model.Mathematics subject classification: 65N06, 65B99.

show abstract

Section: Resultsmentioning

confidence: 99%

Computational Issues in Sensitivity Analysis for 1D Interface Problems

Davis

Singler

2011

JCM

View full text Add to dashboard Cite

show abstract

“…It is well known that a second order accurate approximation to the solution of an interface problem can be generated by the Galerkin finite element method with the standard linear basis functions if the triangulation is aligned with the interface, that is, a body fitted mesh is used (see, e.g., Babuška, 1970;Bramble and King, 1996;Chen and Zou, 1998;Xu, 1982;Karczewska and Boguniewicz, 2016 (Anitescu, 2017), or DPG, a discontinuous Petrov-Galerkin finite element method (Carstensen and Weggler, 2014).…”

Section: Introductionmentioning

confidence: 99%

Accurate gradient computations at interfaces using finite element methods

Qin

Wang

et al. 2017

International Journal of Applied Mathematics and Computer Science

View full text Add to dashboard Cite

New finite element methods are proposed for elliptic interface problems in one and two dimensions. The main motivation is to get not only an accurate solution, but also an accurate first order derivative at the interface (from each side). The key in 1D is to use the idea of Wheeler (1974). For 2D interface problems, the point is to introduce a small tube near the interface and propose the gradient as part of unknowns, which is similar to a mixed finite element method, but only at the interface. Thus the computational cost is just slightly higher than in the standard finite element method. We present a rigorous one dimensional analysis, which shows a second order convergence order for both the solution and the gradient in 1D. For two dimensional problems, we present numerical results and observe second order convergence for the solution, and super-convergence for the gradient at the interface.

show abstract

“…Due to irregular geometry of the boundary and/or interface in many physical phenomena, a natural approach to the numerical approximation is the finite element method (FEM) with unstructured meshes that conform to the geometry of C and oX [2][3][4][5][6][7][8][9]. However, meshing complex interface geometries can prove difficult and time-consuming when the interface frequently changes shape, especially in 3 dimensions.…”

Section: Introductionmentioning

confidence: 99%

“…This algorithm gives more control on the conditioning of the resulting linear system and specifically addresses the conditioning issues (see Appendix C) we found in the straightforward extension of [1] to 3 dimensions. We also give an expanded treatment of the discontinuity removal technique, detailing an algorithm for the construction of a scalar function satisfying the jump conditions (2) and (3). Specific to the 3-dimensional implementation, we describe an algorithm for creating a polyhedral representation of cell-local interface/boundary geometry and quadrature rules suitable for these polyhedral surfaces.…”

Section: Introductionmentioning

confidence: 99%

A second order virtual node method for elliptic problems with interfaces and irregular domains in three dimensions

Hellrung

Wang

Sifakis

et al. 2012

Journal of Computational Physics

View full text Add to dashboard Cite

Keywords: Elliptic interface problems Embedded interface methods Virtual node methods Variational methods Multigrid methods a b s t r a c tWe present a numerical method for the variable coefficient Poisson equation in threedimensional irregular domains and with interfacial discontinuities. The discretization embeds the domain and interface into a uniform Cartesian grid augmented with virtual degrees of freedom to provide accurate treatment of jump and boundary conditions. The matrix associated with the discretization is symmetric positive definite and equal to the standard 7-point finite difference Poisson stencil away from embedded interfaces and boundaries. Numerical evidence suggests second order accuracy in the L 1 -norm. Our approach improves the treatment of Dirichlet and jump constraints in the recent work of Bedrossian et al. [1] and introduces innovations necessary in three dimensions. Specifically, we construct new constraint-based Lagrange multiplier spaces that significantly improve the conditioning of the associated linear system of equations; we provide a method for cell-local polyhedral approximation to the zero isocontour surface of a level set needed for three-dimensional embedding; and we show that the new Lagrange multiplier spaces naturally lead to a class of easy-to-implement multigrid methods that achieve near optimal efficiency, as shown by numerical examples. For the specific case of a continuous Poisson coefficient in interface problems, we provide an expansive treatment of the construction of a particular solution that satisfies the value jump and flux jump constraints. As in [1], this is used in a discontinuity removal technique that yields the standard 7-point stencil across the interface and only requires a modification to the right-hand side of the linear system.

show abstract

A finite element method for interface problems in domains with smooth boundaries and interfaces

Cited by 273 publications

References 14 publications

Computational Issues in Sensitivity Analysis for 1D Interface Problems

Computational Issues in Sensitivity Analysis for 1D Interface Problems

Accurate gradient computations at interfaces using finite element methods

A second order virtual node method for elliptic problems with interfaces and irregular domains in three dimensions

Contact Info

Product

Resources

About