## 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.