椭圆型方程能量积分渐近展开的乘积定理

The Product Theorem of Asymptotic Expansions of the Energy Integral for Elliptic Partial Differential Equations

针对变系数椭圆型方程矩形元,证明了能量积分的渐近展开具有如下的乘积定理:∫Ω∫Ωk2jh2iFi(D2i-2Gj(D2j-2B(w,uh)=∑ny(uyyφ))vhdxdy+ex(uxxφ))vhdxdy+∑nei=1j=1∫Ω∑nh2i[Fij(D2i-2eek2jxD2j-2y(uyyφ))]vhdxdy+Rn,h.xD2jy(uxxφ))

For the elliptic partial differential equations of variable coefficient,we obtain the product theorem of asymptotic expansions of energy integral as follows:B(w,v_h)=∑ni=1h^(2i)_e∫_ΩF_i(D^(2i-2)_x(v_(xx)φ))v_hdxdy+∑nj=1k^(2j)_e∫_ΩG_j(D^(2j-2)_y(u_(yy)φ))u_hdxdy+∑ni+j=2h^(2i)_ek^(2j)_e∫_Ω[F_(ij)(D^(2i-2)_xD^(2j)_y(u_(xx)φ))+G_(ij)(D^(2i)_xD^(2j-2)_y(u_(yy)φ))]v_hdxdy+R_(n,h).