zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
A two-grid finite difference scheme for nonlinear parabolic equations. (English) Zbl 0927.65107

The authors consider the nonlinear parabolic equation

p t-·(K(x,p)p)=f(t,x)in(0,T]×Ω

with the initial condition p(0,x)=p 0 (x) in Ω and Neumann boundary condition (K(x,p)p)·ν=g on (0,T]×Γ, where Ω is a rectangular domain in d (1d3), Γ is its boundary, ν is the outward unit normal vector on Γ and K:Ω× d×d is a symmetric, positive definite second-order diagonal tensor.

Using the variational formulation of the above problem they obtain a two-level nonlinear cell-centered finite difference scheme on a “coarse” grid of mesh size H. For f belonging to L 2 (Ω) on each time-level and for K being of class C 1 the uniqueness and the existence of the solution to the discrete problem is proved, when Δt is small enough, where Δt=max n Δt n , Δt n =t n+1 -t n . An a priori error estimation of order O(H 2 +Δt) is also given. (Here d=2; in general, the order of the error estimation is O(H 4-d|2 +Δt). Next, the authors construct a linear difference scheme on a “fine” grid of mesh size h, hH using the nonlinear solution on the coarse grid. The a priori error estimation is now of order O(H 4-d|2 +h 2 +Δt).

No computational results are given, but they are announced to appear in later papers.

65M06Finite difference methods (IVP of PDE)
35K55Nonlinear parabolic equations
65M12Stability and convergence of numerical methods (IVP of PDE)
65M15Error bounds (IVP of PDE)
65M55Multigrid methods; domain decomposition (IVP of PDE)