We investigate a local modification of a variable-order fractional wave equation, which describes the propagation of diffusive wave in viscoelastic media with evolving physical property. We incorporate an equivalent formulation to prove the well-posedness of the model as well as its high order regularity estimates. To accommodate the convolution term in the reformulated model, we adopt the Ritz-Volterra finite element projection and then derive the rigorous error estimate for the fully-discretized finite element scheme. To circumvent the high computational cost from the temporal integral term, we exploit the translational invariance of the discrete coefficients associated with the convolution structure and construct a fast divide-and-conquer algorithm which reduces the computational complexity from $O(MN^2)$ to $O(MN\log^2 N)$. Numerical experiments are provided to verify the theoretical results and to demonstrate the accuracy and efficiency of the proposed method.

翻译：我们研究了一种变阶分数阶波动方程的局部修正形式，该方程描述了粘弹性介质中物理性质演化时扩散波的传播。通过引入等价形式，我们证明了模型的适定性及其高阶正则性估计。为处理重构模型中的卷积项，我们采用Ritz-Volterra有限元投影方法，进而推导了全离散有限元格式的严格误差估计。为规避时间积分项带来的高计算成本，我们利用卷积结构相关离散系数的平移不变性，构建了一种快速分治算法，将计算复杂度从$O(MN^2)$降低至$O(MN\log^2 N)$。数值实验验证了理论结果，并证明了所提方法的精度与效率。