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...

A Petrov-Galerkin discretization is studied of an ultra-weak variational formulation of the convection-diffusion equation in mixed form. To arrive at an implementable method, the truly optimal test space has to be replaced by its projection onto a finite dimensional test search space. To prevent that this latter space has to be taken increasingly large for vanishing diffusion, a formulation is ...

An accurate and yet simple Meshless Local Petrov-Galerkin (MLPG) formulation for analyzing beam problems is presented. In the formulation, simple weight functions are chosen as test functions. The use of these functions shows that the weak form can be integrated with conventional Gaussian integration. The MLPG method was evaluated by applying the formulation to a variety of patch test and thin ...

We prove convergence and quasi-optimality of a lowest-order adaptive boundary element method for a weakly-singular integral equation in 2D. The adaptive meshrefinement is driven by the weighted-residual error estimator. By proving that this estimator is not only reliable, but under some regularity assumptions on the given data also efficient on locally refined meshes, we characterize the approx...

We present the stability and error analysis of the unified Petrov-Galerkin spectral method, developed in [1], for linear fractional partial differential equations with two-sided derivatives and constant coefficients in any (1 + d)-dimensional space-time hypercube, d = 1, 2, 3, · · · , subject to homogeneous Dirichlet initial/boundary conditions. Specifically, we prove the existence and uniquene...

The truly Meshless Local Petrov-Galerkin (MLPG) method is extended to solve the incompressible Navier-Stokes equations. The local weak form is modified in a very careful way so as to ovecome the so-called Babus̃ka-Brezzi conditions. In addition, The upwinding scheme as developed in Lin and Atluri (2000a) and Lin and Atluri (2000b) is used to stabilize the convection operator in the streamline di...

Abstract. We present the stability and error analysis of the unified Petrov-Galerkin spectral method, developed in [29], for linear fractional partial differential equations with two-sided derivatives and constant coefficients in any (1 + d)-dimensional space-time hypercube, d = 1, 2, 3, · · · , subject to homogeneous Dirichlet initial/boundary conditions. Specifically, we prove the existence a...

We present a generalized polynomial chaos method to solve the steady and unsteady heat transfer problems with uncertainty in boundary conditions, diffusivity coefficient and forcing terms. The stochastic inputs and outputs are represented spectrally by employing the orthogonal polynomial functionals from the Askey scheme, as a generalization of the original polynomial chaos idea of Wiener [1]. ...

Abstract. Non-divergence form elliptic equations with discontinuous coefficients do not generally posses a weak formulation, thus presenting an obstacle to their numerical solution by classical finite element methods. We propose a new hp-version discontinuous Galerkin finite element method for a class of these problems that satisfy the Cordès condition. It is shown that the method exhibits a co...