skip to main content
US FlagAn official website of the United States government
dot gov icon
Official websites use .gov
A .gov website belongs to an official government organization in the United States.
https lock icon
Secure .gov websites use HTTPS
A lock ( lock ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites.

Explore Research Products in the PAR
It may take a few hours for recently added research products to appear in PAR search results.


Title: Geometric transformation of finite element methods: Theory and applications
We present a new technique to apply finite element methods to partial differential equations over curved domains. A change of variables along a coordinate transformation satisfying only low regularity assumptions can translate a Poisson problem over a curved physical domain to a Poisson problem over a polyhedral parametric domain. This greatly simplifies both the geometric setting and the practical implementation, at the cost of having globally rough non-trivial coefficients and data in the parametric Poisson problem. Our main result is that a recently developed broken Bramble-Hilbert lemma is key in harnessing regularity in the physical problem to prove higher-order finite element convergence rates for the parametric problem. Numerical experiments are given which confirm the predictions of our theory.  more » « less
Award ID(s):
2012857
PAR ID:
10524676
Author(s) / Creator(s):
;
Publisher / Repository:
Elsevier
Date Published:
Journal Name:
Applied Numerical Mathematics
Volume:
192
Issue:
C
ISSN:
0168-9274
Page Range / eLocation ID:
389 to 413
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. ; ; (, IMA Journal of Numerical Analysis)
    Abstract In this paper we study the biharmonic equation with Navier boundary conditions in a polygonal domain. In particular, we propose a method that effectively decouples the fourth-order problem as a system of Poisson equations. Our method differs from the naive mixed method that leads to two Poisson problems but only applies to convex domains; our decomposition involves a third Poisson equation to confine the solution in the correct function space, and therefore can be used in both convex and nonconvex domains. A $C^0$ finite element algorithm is in turn proposed to solve the resulting system. In addition, we derive optimal error estimates for the numerical solution on both quasi-uniform meshes and graded meshes. Numerical test results are presented to justify the theoretical findings. 
    more » « less
  2. ; ; ; ; (, ESAIM: Mathematical Modelling and Numerical Analysis)
    We consider an unconstrained tangential Dirichlet boundary control problem for the Stokes equations with an $ L^2 $ penalty on the boundary control.  The contribution of this paper is twofold.  First, we obtain well-posedness and regularity results for the tangential Dirichlet control problem on a convex polygonal domain.  The analysis contains new features not found in similar Dirichlet control problems for the Poisson equation; an interesting result is that the optimal control has higher local regularity on the individual edges of the domain compared to the global regularity on the entire boundary.  Second, we propose and analyze a hybridizable discontinuous Galerkin (HDG) method to approximate the solution.  For convex polygonal domains, our theoretical convergence rate for the control is optimal with respect to the global regularity on the entire boundary.  We present numerical experiments to demonstrate the performance of the HDG method. 
    more » « less
  3. ; ; (, Mathematical Models and Methods in Applied Sciences)
    We obtain regularity results in weighted Sobolev spaces for the solution of the obstacle problem for the integral fractional Laplacian [Formula: see text] in a Lipschitz bounded domain [Formula: see text] satisfying the exterior ball condition. The weight is a power of the distance to the boundary [Formula: see text] of [Formula: see text] that accounts for the singular boundary behavior of the solution for any [Formula: see text]. These bounds then serve us as a guide in the design and analysis of a finite element scheme over graded meshes for any dimension [Formula: see text], which is optimal for [Formula: see text]. 
    more » « less
  4. ; ; ; (, Mathematics of Computation)
    We hybridize the methods of finite element exterior calculus for the Hodge-Laplace problem on differential k-forms in ℝn. In the cases k=0 and k=n, we recover well-known primal and mixed hybrid methods for the scalar Poisson equation, while for 0 
    more » « less
  5. ; (, ESAIM: Mathematical Modelling and Numerical Analysis)
    The paper introduces a new finite element numerical method for the solution of partial differential equations on evolving domains. The approach uses a completely Eulerian description of the domain motion.The physical domain is embedded in a triangulated computational domain and can overlap the time-independent background mesh in an arbitrary way. The numerical method is based on finite difference discretizations of time derivatives and a standard geometrically unfitted finite element method with an additional stabilization term in the spatial domain.The performance and analysis of the method rely on the fundamental extension result in Sobolev spaces for functions defined on bounded domains. This paper includes a complete stability and error analysis, which accounts for discretization errors resulting from finite difference and finite element approximations as well as for geometric errors coming from a possible approximate recovery of the physical domain. Several numerical examples illustrate the theory and demonstrate the practical efficiency of the method. 
    more » « less
  • Citation Formats
  • MLA
  • APA
  • Chicago
  • BibTeX
  • Save / Share this Record
  • Send to Email