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Full and weighted upwind finite element methods. (English) Zbl 0685.65074
Splines in numerical analysis, Contrib. Int. Semin., ISAM-89, Weissig/GDR 1989, Math. Res. 52, 181-188 (1989).
[For the entire collection see Zbl 0664.00022.]
The author considers the singularly perturbed boundary value problem \(- \epsilon u''(x)+b(x)u'(x)=f(x),\) \(u(0)=u(1)=0\) or in a weak formulation: find \(u\in H^ 1_ 0(0,1)\) such that for all \(v\in H_ 0(0,1)\) \(\epsilon (u',v')+(bu',v)=(f,v)\) where (\(\cdot,\cdot)\) denotes the scalar product in \(L^ 2(0,1)\). In using piecewise linear elements on a regular mesh \(0=x_ 0<x_ 1<...<x_ n=1,\) \(h_ i=x_{i+1}-x_ i\) for \(i=0(1)(N-1)\) and assuming that there are positive constants \(c_ 1\), \(c_ 2\) such that \(c_ 1h\leq h_ i\leq c_ 2h\) \((h=\max_{i}h_ i)\) one gets the usual discrete problem: Find \(u_ h\in V_ h\) \((V_ h\) the corresponding, finite element space) such that for all \(v\in V_ h:\) \(\epsilon (u_ h',v')+(bu_ h',v)=(f,v)\).
A secondary decomposition of \((0,1):\quad x_{i+}=(x_{i+1}+x_ i)\) and an elementwise application of the trapezoidal formula for numerical integration of the right hand side leads to the weighted upwind method: find \(u_ h\in V_ h\) such that for all \(v\in V_ h:\) \((*)\quad \epsilon (u_ h',v')+b_ h(u_ h,v)=f_ h(v)\) where \(b_ h\) denotes the bilinear form \(b_ h(u,v)=\sum^{N-1}_{i=0}(u_{i+1}-u_ i)b_{i+}\{\lambda_ iv_{i+1}+\quad (1-\lambda_ i)v_ i\},\) \(u,v\in V_ h\) with upwind parameter \(\lambda_ i\). The author discusses convergence and stability properties of the method (*) in the one- and twodimensional case.
Reviewer: H.Ade

65L10 Numerical solution of boundary value problems involving ordinary differential equations
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
34E15 Singular perturbations for ordinary differential equations
34B05 Linear boundary value problems for ordinary differential equations