Order-reduced analysis and finite element approximation for a viscoelastic membrane equation with Caputo acceleration memory

We investigate the well-posedness, regularity, and numerical approximation of a viscoelastic membrane equation involving a Caputo fractional derivative of order $ \alpha\in(1, 2) $. The fractional term describes an acceleration history memory effect and is therefore different from the velocity memory damping terms commonly used in fractional viscoelastic wave models. By introducing the velocity variable $ z = \partial_tu $ and setting $ \beta = \alpha-1 $, the original problem is reformulated as an order-reduced first-order system in which a Caputo derivative of order $ \beta\in(0, 1) $ acts on the velocity. On the basis of this reformulation, we establish the well-posedness of the initial boundary value problem under homogeneous Neumann or Robin boundary conditions. For the Neumann case, a shifted spectral scale is introduced to include the zero eigenvalue and to obtain regularity estimates uniformly for both the constant and nonconstant modes. We then develop a fully discrete finite element method by combining continuous piecewise linear finite elements in space with the L1 approximation in time. Under suitable regularity and compatibility assumptions, we prove the error estimate

$ \|u-U\|_{\widehat L^\infty(L^2)} + \|\partial_tu-Z\|_{\widehat L^\infty(L^2)} \le C(\tau+h^2). $

Numerical experiments are presented to confirm the predicted first-order temporal convergence and optimal second-order spatial convergence in the $ L^2 $ norm. In addition to the Neumann case, a Robin boundary manufactured solution test is included to demonstrate the applicability of the proposed framework under homogeneous Robin boundary conditions. Additional simulations illustrate the influence of the fractional order on viscoelastic membrane vibration and demonstrate the applicability of the proposed method to modal response analysis.