ABSTRACT

This work aims is to study a nonlinear second-order boundary value differential elliptic problem in one dimension where the nonlinearity concerns the solution and its first derivative. We assume that the source term can be non-smooth and the nonlinearity can grow faster than quadratic. First, we show the existence of a non-negative weak solution if we assume the existence of a super-solution. Second, we present a numerical algorithm to compute an approximation of the non-negative weak solution. The proposed algorithm is decomposed in two steps, the first one is devoted to computing a super-solution, and in the second one, the algorithm computes a sequence of solutions of an intermediate problem obtained by using the Yosida approximation of the nonlinearity. This sequence converges to the non-negative weak solution of the nonlinear equation. The numerical method is an application of the Newton method to the discretized version of the problem, but at each iteration, the resulting system can be indefinite. To overcome this difficulty, we introduce an adaptive non-overlapping domain decomposition method.

摘要

这项工作的目的是研究非线性二阶边值微分椭圆问题，其中非线性涉及解及其一阶导数。我们假设源项可以是非平滑的，并且非线性可以比二次增长更快。首先，如果我们假设存在超解，我们将证明存在非负弱解。其次，我们提出了一种数值算法来计算非负弱解的近似值。所提出的算法分为两步，第一步专门用于计算超解，第二步，该算法计算通过使用非线性的 Yosida 近似获得的中间问题的一系列解。该序列收敛到非线性方程的非负弱解。数值方法是将牛顿方法应用于问题的离散化版本，但在每次迭代中，得到的系统可能是不确定的。为了克服这个困难，我们引入了一种自适应非重叠域分解方法。