论文标题
半线性椭圆PDES的成本最佳自适应迭代线性化FEM
Cost-optimal adaptive iterative linearized FEM for semilinear elliptic PDEs
论文作者
论文摘要
我们考虑标量半线性椭圆形PDE，其中非线性是强烈单调的，但只有局部Lipschitz连续。我们制定了一种自适应迭代线性化有限元法（AILFEM），该方法引导局部网格细化以及出现的非线性离散方程的迭代线性化。为此，我们采用了阻尼的Zarantonello迭代，因此，在算法的每个步骤中，只能求解一个线性泊松型方程。我们证明，提出的AILFEM策略可以保证以最佳速率收敛，在这种情况下，在整体计算复杂性（即计算时间）方面可以理解速率。此外，我们制定并测试一种自适应算法，其中还可以自适应调整Zarantonello迭代的阻尼参数。数值实验强调了理论发现。
We consider scalar semilinear elliptic PDEs where the nonlinearity is strongly monotone, but only locally Lipschitz continuous. We formulate an adaptive iterative linearized finite element method (AILFEM) which steers the local mesh refinement as well as the iterative linearization of the arising nonlinear discrete equations. To this end, we employ a damped Zarantonello iteration so that, in each step of the algorithm, only a linear Poisson-type equation has to be solved. We prove that the proposed AILFEM strategy guarantees convergence with optimal rates, where rates are understood with respect to the overall computational complexity (i.e., the computational time). Moreover, we formulate and test an adaptive algorithm where also the damping parameter of the Zarantonello iteration is adaptively adjusted. Numerical experiments underline the theoretical findings.
