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Abstract In this paper, we study the convergence behavior of the diffuse domain method (DDM) for solving a class of second-order parabolic partial differential equations with Neumann boundary condition posed on general irregular domains. The DDM employs a phase-field function to extend the original parabolic problem to a similar but slightly modified problem defined over a larger rectangular domain that contains the target physical domain. Based on the weighted Sobolev spaces, we rigorously establish the convergence of the diffuse domain solution to the original solution as the interface thickness parameter goes to zero, together with the corresponding optimal error estimates under the weighted $L^2$ and $H^1$ norms. Numerical experiments are also presented to validate the theoretical results.
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This content will become publicly available on May 1, 2027
A Modified Brinkman Penalization Fictitious Domain Method for the Unsteady Navier‐Stokes Equations
ABSTRACT This paper investigates a modification of the fictitious domain method with continuation in the lower‐order coefficients for the unsteady Navier‐Stokes equations governing the motion of an incompressible homogeneous fluid in a bounded 2D or 3D domain. The modification enables a solution‐dependent choice of the critical parameter. Global‐in‐time existence and convergence of a weak solution to the auxiliary problem are proved, and local‐in‐time existence and convergence of a unique strong solution are established. For the strong solution, a new higher‐order convergence rate estimate in the penalization parameter is obtained. The introduced framework allows us to apply a pointwise divergence‐free finite element method as a discretization technique, leading to a strongly mass conservative discrete fictitious domain method. A numerical example illustrates the performance of the method.
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- Award ID(s):
- 2408978
- PAR ID:
- 10686991
- Publisher / Repository:
- Wiley
- Date Published:
- Journal Name:
- Numerical Methods for Partial Differential Equations
- Volume:
- 42
- Issue:
- 3
- ISSN:
- 0749-159X
- Page Range / eLocation ID:
- e70089
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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