Zuodong Wang

Zhaonan Dong, Emmanuil H. Georgoulis, Lorenzo Mascotto et al.

We establish rigorous \emph{a posteriori} error bounds for a space-time finite element method of arbitrary order discretising linear wave problems in second order formulation. The method combines standard finite elements in space and continuous piecewise polynomials in time with an upwind discontinuous Galerkin-type approximation for the second temporal derivative. The proposed scheme accepts dynamic mesh modification, as required by space-time adaptive algorithms, resulting in a discontinuous temporal discretisation when mesh changes occur. We prove \emph{a posteriori} error bounds in the $L^\infty(L^2)$-norm, using carefully designed temporal and spatial reconstructions; explicit control on the constants (including the spatial and temporal orders of the method) in those error bounds is shown. The convergence behaviour of an error estimator is verified numerically, also taking into account the effect of the mesh change. A space-time adaptive algorithm is proposed and tested numerically.

Jie Shen, Zuodong Wang

We propose a high-order finite element method for linear fourth-order elliptic problems that is both nodally bound-preserving and mass-conservative, based on a variational inequality formulation. The method admits an equivalent strictly convex minimization structure, which ensures well-posedness and enables an optimal error estimate in the $H^1$-seminorm, under suitable regularity assumptions. This framework is further extended to nonlinear fourth-order parabolic problems through space--time high-order discretizations that combine variational inequalities, BDF schemes, and scalar auxiliary variable (SAV) techniques. The fully discrete schemes preserve nodal bounds and mass, and a modified energy stability result is established for the first-order temporal scheme. We also apply the same framework to nonlinear second-order parabolic problems by introducing a consistent fourth-order regularization, leading to space--time high-order schemes with the same bound-preserving and mass-conservative properties. Extensive numerical results, including challenging tests with singularities and low regularity, demonstrate the stability, efficiency, and high-order accuracy of the proposed methods.

Zhaonan Dong, Lorenzo Mascotto, Zuodong Wang

We establish fully-discrete a priori and semi-discrete in time a posteriori error estimates for a discontinuous-continuous Galerkin discretization of the wave equation in second order formulation; the resulting method is a Petrov-Galerkin scheme based on piecewise polynomial test functions and continuous piecewise polynomial trial functions in time, respectively. Crucial tools in the a priori analysis for the fully-discrete formulation are the design of suitable projection and interpolation operators extending those used in the parabolic setting, and stability estimates based on a nonstandard choice of the test function; a priori estimates are shown, which are measured in $L^\infty$-type norms in time. For the semi-discrete in time formulation, we exhibit reliable a posteriori error estimates for the error measured in the $L^\infty(L^2)$ norm with fully explicit constants; to this aim, we design a reconstruction operator into $\mathcal C^1$ piecewise polynomials over the time grid with optimal approximation properties in terms of the polynomial degree distribution and the time steps. Numerical examples illustrate the theoretical findings.