51 research outputs found
Maximum Norm Analysis of a Nonmatching Grids Method for Nonlinear Elliptic PDES
We provide a maximum norm analysis of a finite element Schwarz alternating method for a nonlinear elliptic PDE on two overlapping subdomains with nonmatching grids. We consider a domain which is the union of two overlapping subdomains where each subdomain has its own independently generated grid. The two meshes being mutually independent on the overlap region, a triangle belonging to one triangulation does not necessarily belong to the other one. Under a Lipschitz asssumption on the nonlinearity, we establish, on each subdomain, an optimal Lâ error estimate between the discrete Schwarz sequence and the exact solution of the PDE
Singular function mortar finite element methods
This is the published version, also available here: http://dx.doi.org/10.2478/cmam-2003-0014.We consider the Poisson equation with Dirichlet boundary conditions on a polygonal domain with one reentrant corner. We introduce new nonconforming finite element discretizations based on mortar techniques and singular functions. The main idea introduced in this paper is the replacement of cut-off functions by mortar element techniques on the boundary of the domain. As advantages, the new discretizations do not require costly numerical integrations and have smaller a priori error estimates and condition numbers. Based on such an approach, we prove optimal accuracy error bounds for the discrete solution. Based on such techniques, we also derive new extraction formulas for the stress intensive factor. We establish optimal accuracy for the computed stress intensive factor. Numerical examples are presented to support our theory
Robust a posteriori error control and adaptivity for multiscale, multinumerics, and mortar coupling
International audienceWe consider discretizations of a model elliptic problem by means of different numerical methods applied separately in different subdomains, termed multinumerics, coupled using the mortar technique. The grids need not match along the interfaces. We are also interested in the multiscale setting, where the subdomains are partitioned by a mesh of size h , whereas the interfaces are partitioned by a mesh of much coarser size H , and where lower-order polynomials are used in the subdomains and higher-order polynomials are used on the mortar interface mesh. We derive several fully computable a posteriori error estimates which deliver a guaranteed upper bound on the error measured in the energy norm. Our estimates are also locally efficient and one of them is robust with respect to the ratio H/h under an assumption of sufficient regularity of the weak solution. The present approach allows bounding separately and comparing mutually the subdomain and interface errors. A subdomain/interface adaptive refinement strategy is proposed and numerically tested
Nonoverlapping domain decomposition preconditioners for discontinuous Galerkin approximations of Hamilton--Jacobi--Bellman equations
We analyse a class of nonoverlapping domain decomposition preconditioners for
nonsymmetric linear systems arising from discontinuous Galerkin finite element
approximation of fully nonlinear Hamilton--Jacobi--Bellman (HJB) partial
differential equations. These nonsymmetric linear systems are uniformly bounded
and coercive with respect to a related symmetric bilinear form, that is
associated to a matrix . In this work, we construct a
nonoverlapping domain decomposition preconditioner , that is based
on , and we then show that the effectiveness of the preconditioner
for solving the} nonsymmetric problems can be studied in terms of the condition
number . In particular, we establish the
bound , where
and are respectively the coarse and fine mesh sizes, and and
are respectively the coarse and fine mesh polynomial degrees. This represents
the first such result for this class of methods that explicitly accounts for
the dependence of the condition number on ; our analysis is founded upon an
original optimal order approximation result between fine and coarse
discontinuous finite element spaces. Numerical experiments demonstrate the
sharpness of this bound. Although the preconditioners are not robust with
respect to the polynomial degree, our bounds quantify the effect of the coarse
and fine space polynomial degrees. Furthermore, we show computationally that
these methods are effective in practical applications to nonsymmetric, fully
nonlinear HJB equations under -refinement for moderate polynomial degrees
Space-time domain decomposition for advection-diffusion problems in mixed formulations
This paper is concerned with the numerical solution of porous-media flow and
transport problems , i. e. heterogeneous, advection-diffusion problems. Its aim
is to investigate numerical schemes for these problems in which different time
steps can be used in different parts of the domain. Global-in-time,
non-overlapping domain-decomposition methods are coupled with operator
splitting making possible the different treatment of the advection and
diffusion terms. Two domain-decomposition methods are considered: one uses the
time-dependent Steklov--Poincar{\'e} operator and the other uses optimized
Schwarz waveform relaxation (OSWR) based on Robin transmission conditions. For
each method, a mixed formulation of an interface problem on the space-time
interface is derived, and different time grids are employed to adapt to
different time scales in the subdomains. A generalized Neumann-Neumann
preconditioner is proposed for the first method. To illustrate the two methods
numerical results for two-dimensional problems with strong heterogeneities are
presented. These include both academic problems and more realistic prototypes
for simulations for the underground storage of nuclear waste
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