Error analysis of finite element solutions to nonlinear partial differential equations

Let？The following strongly nonlinear elliptic boundary value proManagement has been considered ：イD2（aイD4 ￣イエD4（·，x，u，·u）·让F（λ，u）是由上面等式定义的非线性运算符。We have shown， using the Kantorovich theorem and the Implicit Function Theorem with error estimation， that if （λ，u） is an exact solution of the equation and the Frechet derivative D2uie D2F（λ，u） with respect to u is an isomorphism between certain function spaces then there exists a locally unique finite element solution （λ，u） closed to （λ，u） and several error estimates are obtained。这个结果可以在几种方式中扩展。即使解决方案分支也有转折点，我们也可以获得类似的结果。在这样的情况下，finite element solution的错误（|\-\D2h\D2| + ||u-u-i|| <_C|| u-イD2hエD2u||好吧，我们可以展示这个错误。|\-\D2h\D2|这个错误太小了||u-u-i||.如果这个等式有一个邀请条款，那么我们就应该引入这样一个称为向上的有限元素计划，以获得更好的近似。However，这一类的鉴别模具是一个非差别的有限元素运算符。Even so， we can obtain similar error analysis if the discirtized operator has a "pseudo-derivative"。

Let Ω ⊂ RィイD1dィエD1 be a bounded domain in the d-dimensional Euclidean space RィイD1dィエD1. The following strongly nonlinear elliptic boundary value problem has been considered :∫ィイD2ΩィエD2(aィイD4→ィエD4(λ,x,u,∇u)・∇ν+f(λ,x,u,∇u)ν)=0, ∀ν ∈ ΗィイD31(/)0ィエD3(Ω),where aィイD4→ィエD4, f are sufficiently smooth functions. Let F(λ,u) be the nonlinear operator defined by the above equation. We have shown, using the Kantorovich theorem and the Implicit Function Theorem with error estimation, that if (λ,u) is an exact solution of the equation and the Frechet derivative DィイD2uィエD2F(λ,u) with respect to u is an isomorphism between certain function spaces then there exists a locally unique finite element solution (λ,uィイD2hィエD2) closed to (λ,u) and several error estimates are obtained. This result can be extended in a few ways. Even if solution branch has turning points we can obtain similar results. In such a case, the error of the finite element solution (λィイD2hィエD2,uィイD2hィエD2) is estimated as|λ-λィイD2hィエD2|+ ||u-uィイD2hィエD2||<_C||u-ΠィイD2hィエD2u||.Moreover, we can show that the error |λ-λィイD2hィエD2| is much smaller that the error ||u-uィイD2hィエD2||. If the equation has a convection term, we have to introduce so-called upwind finite element scheme to obtain better approximation. However, such kind of discritization yields a non-differentiable finite element operator. Even so, we can obtain similar error analysis if the discirtized operator has a "pseudo-derivative".