ISSN:
1572-9044
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract In this paper, we establish some error bounds for the continuous piecewise linear finite element approximation of the following problem: Let Ω be an open set in ℝ d , withd=1 or 2. GivenT〉0,p ∈ (1, ∞),f andu 0; findu ∈K, whereK is a closed convex subset of the Sobolev spaceW 0 1,p (Ω), such that for anyv∈K $$\begin{gathered} \int\limits_\Omega {u_1 (\upsilon - u) dx + } \int\limits_\Omega {\left| {\nabla u} \right|^{p - 2} } \nabla u \cdot \nabla (\upsilon - u) dx \geqslant \int\limits_\Omega {f(\upsilon - u) dx for} a.e. t \in (0,T], \hfill \\ u = 0 on \partial \Omega \times (0,T] and u(0,x) = u_0 (x) for x \in \Omega . \hfill \\ \end{gathered} $$ We prove error bounds in energy type norms for the fully discrete approximation using the backward Euler time discretisation. In some notable cases, these error bounds converge at the optimal rate with respect to the space discretisation, provided the solutionu is sufficiently regular.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF02071387
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