In this paper, we present a two-grid scheme for a semilinear parabolic integro-differential equation using a new mixed finite element method. The gradient for the method belongs to the square integrable space instead of the classical H(div; Ω) space. The velocity and the pressure are approximated by the P₀²–P₁ pair which satisfies the inf-sup condition. Firstly, we solve an original nonlinear problem on the coarse grid in our two-grid scheme. Then, to linearize the discretized equations, we use Newton iteration on the fine grid twice. It is shown that the algorithm can achieve asymptotically optimal approximation as long as the mesh sizes satisfy h = O(H⁶|lnH|²). As a result, solving such a large class of nonlinear equations will not be much more difficult than the solution of one linearized equation. Finally, a numerical experiment is provided to verify theoretical results of the two-grid method.

中文翻译：

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半线性抛物型积分微分方程混合有限元逼近的两网格方法

在本文中，我们提出了一种使用新的混合有限元方法的半线性抛物型积分微分方程的两重网格格式。该方法的梯度属于正方形可积空间，而不是经典H（div;Ω ）空间。速度和压力由满足inf-up条件的P ₀ ² –P ₁对近似。首先，我们在两网格方案中解决了粗网格上的原始非线性问题。然后，为了线性化离散方程，我们在细网格上两次使用牛顿迭代。结果表明，只要网格尺寸满足h = O（H ⁶，则该算法就可以实现渐近最优逼近。| ln H | ²）。结果，解决这样一大类非线性方程不会比一个线性方程的求解困难得多。最后，通过数值实验验证了两网格法的理论结果。