Analysis of the diffuse-domain method for solving PDEs in complex geometries

Lervåg, Karl Yngve; Lowengrub, John

doi:10.4310/cms.2015.v13.n6.a6

Cited by 34 publications

(33 citation statements)

References 69 publications

(73 reference statements)

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“…Therefore, a lot of research has been conducted to construct accurate methods employing meshes which are not aligned with Γ but possess a "simple" structure and are fixed throughout the simulation; see for instance the immersed boundary method [42], the immersed interface method [33,37], immersed finite elements [36,39,45], the fictitious domain method [5,27,28,40], the unfitted finite element method [7,19,31,34], the finite cell method [41], unfitted discontinuous Galerkin methods [9], or composite finite elements [30,38], and the references provided there. In this work we will focus on a diffuse interface method for solving (1), see for instance [1,17,23,24,32,35,43]. In this method the sharp interface condition u = g on Γ is replaced by suitable conditions on u − g on a diffuse layer centered around Γ.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Error analysis of a diffuse interface method for elliptic problems with Dirichlet boundary conditions

Schlottbom

2016

Applied Numerical Mathematics

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Abstract. We use a diffuse interface method for solving Poisson's equation with a Dirichlet condition on an embedded curved interface. The resulting diffuse interface problem is identified as a standard Dirichlet problem on approximating regular domains. We estimate the errors introduced by these domain perturbations, and prove convergence and convergence rates in the H 1 -norm, the L 2 -norm and the L ∞ -norm in terms of the width of the diffuse layer. For an efficient numerical solution we consider the finite element method for which another domain perturbation is introduced. These perturbed domains are polygonal and non-convex in general. We prove convergence and convergences rates in the H 1 -norm and the L 2 -norm in terms of the layer width and the mesh size. In particular, for the L 2 -norm estimates we present a problem adapted duality technique, which crucially makes use of the error estimates derived for the regularly perturbed domains. Our results are illustrated by numerical experiments, which also show that the derived estimates are sharp.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Error analysis of a diffuse interface method for elliptic problems with Dirichlet boundary conditions

Schlottbom

2016

Applied Numerical Mathematics

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“…Note that for DDM3, since the phase field function φ and its gradient vanish rapidly outside D, in order to prevent the equation from being ill-posed, we use the following modified gradient instead in the numerical calculation [75],…”

Section: Diffuse Domain Methods To the Poisson Equation With Dirichletmentioning

confidence: 99%

“…Using rigorous mathematical theory [71,72,73,74], matched asymptotic expansions and numerical simulations (e.g., [45,46,47,48,49]), the DDMs have been shown to converge to the original PDE and boundary conditions as the diffuse interface parameter tends to zero. Further, in [75] a matched asymptotic analysis for general DDMs with Neumann and Robin boundary conditions showed that for certain choices of the source terms, the DDMs were secondorder accurate in and in the grid size h in both the L 2 and L ∞ norms, taking ∝ h; see the recent paper [74] for a rigorous proof.…”

Section: Introductionmentioning

confidence: 98%

Higher-order accurate diffuse-domain methods for partial differential equations with Dirichlet boundary conditions in complex, evolving geometries

Guo

Lowengrub

2020

Journal of Computational Physics

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The diffuse-domain, or smoothed boundary, method is an attractive approach for solving partial differential equations in complex geometries because of its simplicity and flexibility. In this method the complex geometry is embedded into a larger, regular domain. The original PDE is reformulated using a smoothed characteristic function of the complex domain and source terms are introduced to approximate the boundary conditions. The reformulated equation, which is independent of the dimension and domain geometry, can be solved by standard numerical methods and the same solver can be used for any domain geometry. A challenge is making the method higher-order accurate. For Dirichlet boundary conditions, which we focus on here, current implementations demonstrate a wide range in their accuracy but generally the methods yield at best first order accuracy in , the parameter that characterizes the width of the region over which the characteristic function is smoothed. Typically, ∝ h, the grid size. Here, we analyze the diffuse-domain PDEs using matched asymptotic expansions and explain the observed behaviors. Our analysis also identifies simple modifications to the diffuse-domain PDEs that yield higher-order accuracy in , e.g., O( 2 ) in the L 2 norm and O( p ) with 1.5 ≤ p ≤ 2 in the L ∞ norm. Our analytic results are confirmed numerically in stationary and moving domains where the level set method is used to capture the dynamics of the domain boundary and to construct the smoothed characteristic function.

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“…Finite element methods based on diffuse boundaries,() also known as diffuse domain, phase‐field, fat boundary, or spread interface methods, provide a pathway for solving boundary value problems on very complex domains without the need for explicitly parameterizing boundary and interface surfaces. Various instantiations related to the phase‐field concept have been published in the last few years, eg, for advection‐diffusion problems,() multiphase flow,() the evolution of complex crack patterns,() fluid infiltration and biomedical growth processes,() and phase transition and segregation processes.…”

Section: Introductionmentioning

confidence: 99%

The diffuse Nitsche method: Dirichlet constraints on phase‐field boundaries

Nguyen

Stoter

Ruess

et al. 2017

Numerical Meth Engineering

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Summary We explore diffuse formulations of Nitsche's method for consistently imposing Dirichlet boundary conditions on phase‐field approximations of sharp domains. Leveraging the properties of the phase‐field gradient, we derive the variational formulation of the diffuse Nitsche method by transferring all integrals associated with the Dirichlet boundary from a geometrically sharp surface format in the standard Nitsche method to a geometrically diffuse volumetric format. We also derive conditions for the stability of the discrete system and formulate a diffuse local eigenvalue problem, from which the stabilization parameter can be estimated automatically in each element. We advertise metastable phase‐field solutions of the Allen‐Cahn problem for transferring complex imaging data into diffuse geometric models. In particular, we discuss the use of mixed meshes, that is, an adaptively refined mesh for the phase‐field in the diffuse boundary region and a uniform mesh for the representation of the physics‐based solution fields. We illustrate accuracy and convergence properties of the diffuse Nitsche method and demonstrate its advantages over diffuse penalty‐type methods. In the context of imaging‐based analysis, we show that the diffuse Nitsche method achieves the same accuracy as the standard Nitsche method with sharp surfaces, if the inherent length scales, ie, the interface width of the phase‐field, the voxel spacing, and the mesh size, are properly related. We demonstrate the flexibility of the new method by analyzing stresses in a human vertebral body.

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Analysis of the diffuse-domain method for solving PDEs in complex geometries

Cited by 34 publications

References 69 publications

Error analysis of a diffuse interface method for elliptic problems with Dirichlet boundary conditions

Error analysis of a diffuse interface method for elliptic problems with Dirichlet boundary conditions

Higher-order accurate diffuse-domain methods for partial differential equations with Dirichlet boundary conditions in complex, evolving geometries

The diffuse Nitsche method: Dirichlet constraints on phase‐field boundaries

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