Ultraweak formulation of linear PDEs in nondivergence form and DPG approximation

An ultraweak formulation of the Reissner-Mindlin plate bending model and DPG approximation

We develop and analyze an ultraweak variational formulation of the Reissner-Mindlin plate bending model both for the clamped and the soft simply supported cases. We prove well-posedness of the formulation, uniformly with respect to the plate thickness $t$. We also prove weak convergence of the Reissner-Mindlin solution to the solution of the corresponding Kirchhoff-Love model when $t\to 0$. Based on the ultraweak formulation, we introduce a discretization of the discontinuous Petrov-Galerkin type with optimal test functions (DPG) and prove its uniform quasi-optimal convergence. Our theory cove…

Kirchhoff-Love板弯曲模型的超弱形式及其DPG近似

An ultraweak formulation of the Kirchhoff-Love plate bending model and DPG approximation

本文提出Kirchhoff-Love板弯曲模型的超弱变分形式，并基于该形式引入最优测试函数的断续Petrov-Galerkin型离散化方法，证明了超弱形式的适定性和DPG方案的准最优收敛性。

一个用于薛定谔方程的空间时间DPG方法

A spacetime DPG method for the Schrodinger equation

本文提出了一种用于线性时间依赖薛定谔方程的空间时间不连续Petrov-Galerkin (DPG)方法，通过使用二阶薛定谔算子来捕捉不规则解，并分析了基于弱形式的DPG方法的收敛性。

一种用于粘弹性流体的超弱DPG方法

An ultraweak DPG method for viscoelastic fluids

本文提出了一种超弱DPG方法用于粘弹性流体流动的基准问题，通过自适应网格生成实现后验误差指标，并利用对称正定刚度矩阵提高求解效率。

A DPG method for shallow shells

We develop and analyze a discontinuous Petrov--Galerkin method with optimal test functions (DPG method) for a shallow shell model of Koiter type. It is based on a uniformly stable ultraweak formulation and thus converges robustly quasi-uniformly. Numerical experiments for various cases, including the Scordelis--Lo cylindrical roof, elliptic and hyperbolic geometries, illustrate its performance. The built-in DPG error estimator gives rise to adaptive mesh refinements that are capable to resolve boundary and interior layers. The membrane locking is dealt with by raising the polynomial degree onl…

A DPG method for the quad-curl problem

We derive an ultraweak variational formulation of the quad-curl problem in two and three dimensions. We present a discontinuous Petrov-Galerkin (DPG) method for its approximation and prove its quasi-optimal convergence. We illustrate how this method can be applied to the Stokes problem in two dimensions, after an application of the curl operator to eliminate the pressure variable. In this way, DPG techniques known from Kirchhoff-Love plates can be used. We present an a priori error estimate that improves a previous approximation result for effective shear forces by using a less restrictive reg…