Geodesic
Effective difference schemes for the heat equation in arbitrary regions
Mathematica Applicanda, Tome 10 (1982) no. 19, pp. 5-13.

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In this paper the author considers the problem of the heat equation ∂u/∂t−(∂2u/∂x21+∂2u/∂x22)=f(x,t) for x∈Ω and t∈(0,T], u(x,0)=φ(x) for x∈Ω, u(x,t)=0 for x∈∂Ω and t∈[0,T]. He constructs a Crank-Nicolson and an alternating direction difference scheme on a regular mesh with steps hi (i=1,2) and τ. Linear interpolation is used for the approximation of the boundary condition. Besides stability of both schemes error estimates are derived under the condition that the derivatives ∂5u/∂t∂x4i and ∂3u/∂t3 are bounded. These estimates are: maxn∥un−yn∥A≤M(τ2+h3/2)andmaxn∥un−yn∥h≤M(τ2+h2+τh1/2+h5/2/τ). Here h=max(h1,h2), un=u(⋅,nτ), yn is the approximate value of un, ∥u∥2h=(u,u)h, (u,v)h=h1h2∑x∈Ωhu(x)v(x) (Ωh is the set of all mesh points lying in Ω), and ∥u∥2A=(u,Au)h where A is the discrete Laplace operator.
DOI : 10.14708/ma.v10i19.1529
Classification : 65N10(65N15)
Mots-clés : Stability and convergence of difference methods,Error bounds 
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     title = {Effective difference schemes for the heat equation in arbitrary regions},
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Maksymilian Dryja. Effective difference schemes for the heat equation in arbitrary regions. Mathematica Applicanda, Tome 10 (1982) no. 19, pp.  5-13. doi : 10.14708/ma.v10i19.1529. http://geodesic.mathdoc.fr/articles/10.14708/ma.v10i19.1529/

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