The standard a priori error analysis of discontinuous Galerkin methods requires additional regularity on the solution of the elliptic boundary value problem in order to justify the Galerkin orthogonality and to handle the normal derivative on element interfaces that appear in the discrete energy norm. In this paper, a new error analysis of discontinuous Galerkin methods is developed using only ...

Three new Eulerian finite element methods for parabolic PDEs on a moving surface Γ(t) are presented and compared in numerical experiments. These are space-time Galerkin methods, which are derived from a weak formulation in space and time. The trialand test-spaces contain the traces on the space-time manifold of an outer prismatic finite element space. The numerical experiments show that two of ...

In this paper, a stochastic finite element approximation scheme is developed for an optimal control problem governed by an elliptic integro-differential equation with stochastic coefficients. Different from the well-studied optimal control problems governed by stochastic PDEs, our control problem has the control constraints of obstacle type, which is mostly seen in real applications. We develop...

A Discontinuous Galerkin (DG) method with solenoidal approximation for the simulation of incompressible flow is proposed. It is applied to the solution of the Stokes equations. The Interior Penalty Method is employed to construct the DG weak form. For every element, the approximation space for the velocity field is decomposed as direct sum of a solenoidal space and an irrotational space. This a...

In this paper, a notion of invariant Galerkin-variational weak forms is proposed. Two speci c invariant variational weak forms, the J -invariant and the L-invariant, are constructed based on the corresponding conservation laws in elasticity, one of which is the conservation of Eshelby’s energy-momentum (Eshelby, Philos. Trans. Roy. Soc. 1951; 87:12; In Solid State Physics, Setitz F, Turnbull D ...

A meshfree radial point interpolation method (RPIM) is developed for stress analysis of threedimensional (3D) solids, based on the Galerkin weak form formulation using 3D meshfree shape functions constructed using radial basis functions (RBFs). As the RPIM shape functions have the Kronecker delta functions property, essential boundary conditions can be enforced as easily as in the finite elemen...