39 research outputs found
The BEM with graded meshes for the electric field integral equation on polyhedral surfaces
We consider the variational formulation of the electric field integral
equation on a Lipschitz polyhedral surface . We study the Galerkin
boundary element discretisations based on the lowest-order Raviart-Thomas
surface elements on a sequence of anisotropic meshes algebraically graded
towards the edges of . We establish quasi-optimal convergence of
Galerkin solutions under a mild restriction on the strength of grading. The key
ingredient of our convergence analysis are new componentwise stability
properties of the Raviart-Thomas interpolant on anisotropic elements
A posteriori error estimation and adaptivity in stochastic Galerkin FEM for parametric elliptic PDEs: beyond the affine case
We consider a linear elliptic partial differential equation (PDE) with a
generic uniformly bounded parametric coefficient. The solution to this PDE
problem is approximated in the framework of stochastic Galerkin finite element
methods. We perform a posteriori error analysis of Galerkin approximations and
derive a reliable and efficient estimate for the energy error in these
approximations. Practical versions of this error estimate are discussed and
tested numerically for a model problem with non-affine parametric
representation of the coefficient. Furthermore, we use the error reduction
indicators derived from spatial and parametric error estimators to guide an
adaptive solution algorithm for the given parametric PDE problem. The
performance of the adaptive algorithm is tested numerically for model problems
with two different non-affine parametric representations of the coefficient.Comment: 32 pages, 4 figures, 6 table
A posteriori error estimation and adaptivity in stochastic Galerkin FEM for parametric elliptic PDEs: beyond the affine case
We consider a linear elliptic partial differential equation (PDE) with a
generic uniformly bounded parametric coefficient. The solution to this PDE
problem is approximated in the framework of stochastic Galerkin finite element
methods. We perform a posteriori error analysis of Galerkin approximations and
derive a reliable and efficient estimate for the energy error in these
approximations. Practical versions of this error estimate are discussed and
tested numerically for a model problem with non-affine parametric
representation of the coefficient. Furthermore, we use the error reduction
indicators derived from spatial and parametric error estimators to guide an
adaptive solution algorithm for the given parametric PDE problem. The
performance of the adaptive algorithm is tested numerically for model problems
with two different non-affine parametric representations of the coefficient.Comment: 32 pages, 4 figures, 6 table
Adaptive FEM with coarse initial mesh guarantees optimal convergence rates for compactly perturbed elliptic problems
We prove that for compactly perturbed elliptic problems, where the
corresponding bilinear form satisfies a Garding inequality, adaptive
mesh-refinement is capable of overcoming the preasymptotic behavior and
eventually leads to convergence with optimal algebraic rates. As an important
consequence of our analysis, one does not have to deal with the a-priori
assumption that the underlying meshes are sufficiently fine. Hence, the overall
conclusion of our results is that adaptivity has stabilizing effects and can
overcome possibly pessimistic restrictions on the meshes. In particular, our
analysis covers adaptive mesh-refinement for the finite element discretization
of the Helmholtz equation from where our interest originated
Adaptive BEM with optimal convergence rates for the Helmholtz equation
We analyze an adaptive boundary element method for the weakly-singular and
hypersingular integral equations for the 2D and 3D Helmholtz problem. The
proposed adaptive algorithm is steered by a residual error estimator and does
not rely on any a priori information that the underlying meshes are
sufficiently fine. We prove convergence of the error estimator with optimal
algebraic rates, independently of the (coarse) initial mesh. As a technical
contribution, we prove certain local inverse-type estimates for the boundary
integral operators associated with the Helmholtz equation
Convergence of adaptive stochastic Galerkin FEM
We propose and analyze novel adaptive algorithms for the numerical solution
of elliptic partial differential equations with parametric uncertainty. Four
different marking strategies are employed for refinement of stochastic Galerkin
finite element approximations. The algorithms are driven by the energy error
reduction estimates derived from two-level a posteriori error indicators for
spatial approximations and hierarchical a posteriori error indicators for
parametric approximations. The focus of this work is on the mathematical
foundation of the adaptive algorithms in the sense of rigorous convergence
analysis. In particular, we prove that the proposed algorithms drive the
underlying energy error estimates to zero
Goal-oriented adaptivity for multilevel stochastic Galerkin FEM with nonlinear goal functionals
This paper is concerned with the numerical approximation of quantities of
interest associated with solutions to parametric elliptic partial differential
equations (PDEs). The key novelty of this work is in its focus on the
quantities of interest represented by continuously G\^ateaux differentiable
nonlinear functionals. We consider a class of parametric elliptic PDEs where
the underlying differential operator has affine dependence on a countably
infinite number of uncertain parameters. We design a goal-oriented adaptive
algorithm for approximating nonlinear functionals of solutions to this class of
parametric PDEs. In the algorithm, the approximations of parametric solutions
to the primal and dual problems are computed using the multilevel stochastic
Galerkin finite element method (SGFEM) and the adaptive refinement process is
guided by reliable spatial and parametric error indicators that identify the
dominant sources of error. We prove that the proposed algorithm generates
multilevel SGFEM approximations for which the estimates of the error in the
goal functional converge to zero. Numerical experiments for a selection of test
problems and nonlinear quantities of interest demonstrate that the proposed
goal-oriented adaptive strategy yields optimal convergence rates (for both the
error estimates and the reference errors in the quantities of interest) with
respect to the overall dimension of the underlying multilevel approximations
spaces.Comment: 26 pages, 2 figure