Dual weighted residual based error control for nonstationary convection-dominated equations: potential or ballast?

Even though substantial progress has been made in the numerical approximation of convection-dominated problems, its major challenges remain in the scope of current research. In particular, parameter robust a posteriori error estimates for quantities of physical interest and adaptive mesh refinement strategies with proved convergence are still missing. Here, we study numerically the potential of the Dual Weighted Residual (DWR) approach applied to stabilized finite element methods to further enhance the quality of approximations. The impact of a strict application of the DWR methodology is particularly focused rather than the reduction of computational costs for solving the dual problem by interpolation or localization.Comment: arXiv admin note: text overlap with arXiv:1803.1064

Primal dual mixed finite element methods for indefinite advection--diffusion equations

We consider primal-dual mixed finite element methods for the advection--diffusion equation. For the primal variable we use standard continuous finite element space and for the flux we use the Raviart-Thomas space. We prove optimal a priori error estimates in the energy- and the $L^2$-norms for the primal variable in the low Peclet regime. In the high Peclet regime we also prove optimal error estimates for the primal variable in the $H(div)$ norm for smooth solutions. Numerically we observe that the method eliminates the spurious oscillations close to interior layers that pollute the solution of the standard Galerkin method when the local Peclet number is high. This method, however, does produce spurious solutions when outflow boundary layer presents. In the last section we propose two simple strategies to remove such numerical artefacts caused by the outflow boundary layer and validate them numerically.Comment: 25 pages, 6 figures, 5 table

Space Decompositions and Solvers for Discontinuous Galerkin Methods

We present a brief overview of the different domain and space decomposition techniques that enter in developing and analyzing solvers for discontinuous Galerkin methods. Emphasis is given to the novel and distinct features that arise when considering DG discretizations over conforming methods. Connections and differences with the conforming approaches are emphasized.Comment: 2 pages 2 figures no table

A subgrid viscosity Lagrance-Galerkin method for convection-diffusion problems

We present and analyze a subgrid viscosity Lagrange-Galerk in method that combines the subgrid eddy viscosity method proposed in W. Layton, A connection between subgrid scale eddy viscosity and mixed methods. Appl. Math. Comp., 133: 14 7-157, 2002, and a conventional Lagrange-Galerkin method in the framework of P1⊕ cubic bubble finite elements. This results in an efficient and easy to implement stabilized method for convection dominated convection diffusion reaction problems. Numerical experiments support the numerical analysis results and show that the new method is more accurate than the conventional Lagrange-Galerkin one

Numerical Analysis of a Second Order Pure Lagrange--Galerkin Method for Convection-Diffusion Problems. Part II: Fully Discretized Scheme and Numerical Results

This version of the article has been accepted for publication, after peer review, but is not the Version of Record and does not reflect post-acceptance improvements, or any corrections. The Version of Record is available online at: https://doi.org/10.1137/100809994[Abstract]: We analyze a second order pure Lagrange-Galerkin method for variable coefficient convection-(possibly degenerate) diffusion equations with mixed Dirichlet-Robin boundary conditions. In a previous paper the proposed second order pure Lagrangian time discretization scheme has been introduced and analyzed for the same problem. More precisely, the l1(H1) stability and l1(H1) error estimates of order O(_t2) has been obtained. Moreover, for the particular case of incompressible flows, stability inequalities with constants independent of the final time have been stated. In the present paper l1(H1) error estimates of order O(_t2) + O(hk) are obtained for the fully discretized pure Lagrange-Galerkin method. To prove these results we use some properties obtained in the previous paper. Finally, numerical tests are presented that confirm the theoretical results.This work was supported by Xunta de Galicia under research project INCITE09 207 047 PR, and by Ministerio de Ciencia e Innovación (Spain) under research projects Consolider MATHEMATICA CSD2006-00032 and MTM2008-02483. Xunta de Galicia; INCITE09 207 047 P

A first order system least squares method for the Helmholtz equation

We present a first order system least squares (FOSLS) method for the Helmholtz equation at high wave number k, which always deduces Hermitian positive definite algebraic system. By utilizing a non-trivial solution decomposition to the dual FOSLS problem which is quite different from that of standard finite element method, we give error analysis to the hp-version of the FOSLS method where the dependence on the mesh size h, the approximation order p, and the wave number k is given explicitly. In particular, under some assumption of the boundary of the domain, the L2 norm error estimate of the scalar solution from the FOSLS method is shown to be quasi optimal under the condition that kh/p is sufficiently small and the polynomial degree p is at least O(\log k). Numerical experiments are given to verify the theoretical results