Quasi-uniform Meshes Research Articles

Analysis of discontinuous Galerkin methods using mesh-dependent norms and applications to problems with rough data

We prove the inf-sup stability of a discontinuous Galerkin scheme for second order elliptic operators in (unbalanced) mesh-dependent norms for quasi-uniform meshes for all spatial dimensions. This results in a priori error bounds in these norms. As an application we examine some problems with rough source term where the solution can not be characterised as a weak solution and show quasi-optimal error control.

Local error estimates of the finite element method for an elliptic problem with a Dirac source term

The solutions of elliptic problems with a Dirac measure right‐hand side are not in dimension and therefore the convergence of the finite element solutions is suboptimal in the ‐norm. In this article, we address the numerical analysis of the finite element method for the Laplace equation with Dirac source term: we consider, in dimension 3, the Dirac measure along a curve and, in dimension 2, the punctual Dirac measure. The study of this problem is motivated by the use of the Dirac measure as a reduced model in physical problems, for which high accuracy of the finite element method at the singularity is not required. We show a quasioptimal convergence in the ‐norm, for on subdomains which exclude the singularity; in the particular case of Lagrange finite elements, an optimal convergence in ‐norm is shown on a family of quasiuniform meshes. Our results are obtained using local Nitsche and Schatz‐type error estimates, a weak version of Aubin‐Nitsche duality lemma and a discrete inf‐sup condition. These theoretical results are confirmed by numerical illustrations.

Aspects of an adaptive finite element method for the fractional Laplacian: A priori and a posteriori error estimates, efficient implementation and multigrid solver

We develop all of the components needed to construct an adaptive finite element code that can be used to approximate fractional partial differential equations, on non-trivial domains in d≥1 dimensions. Our main approach consists of taking tools that have been shown to be effective for adaptive boundary element methods and, where necessary, modifying them so that they can be applied to the fractional PDE case. Improved a priori error estimates are derived for the case of quasi-uniform meshes which are seen to deliver sub-optimal rates of convergence owing to the presence of singularities. Attention is then turned to the development of an a posteriori error estimate and error indicators which are suitable for driving an adaptive refinement procedure. We assume that the resulting refined meshes are locally quasi-uniform and develop efficient methods for the assembly of the resulting linear algebraic systems and their solution using iterative methods, including the multigrid method. The storage of the dense matrices along with efficient techniques for computing the dense matrix–vector products needed for the iterative solution is also considered. The performance and efficiency of the resulting algorithm is illustrated for a variety of examples.

Superconvergence of Discontinuous Galerkin methods based on upwind-biased fluxes for 1D linear hyperbolic equations

In this paper, we study superconvergence properties of the discontinuous Galerkin method using upwind-biased numerical fluxes for one-dimensional linear hyperbolic equations. A (2 k + 1)th order superconvergence rate of the DG approximation at the numerical fluxes and for the cell average is obtained under quasi-uniform meshes and some suitable initial discretization, when piecewise polynomials of degree k are used. Furthermore, surprisingly, we find that the derivative and function value approximation of the DG solution are superconvergent at a class of special points, with an order k + 1 and k + 2, respectively. These superconvergent points can be regarded as the generalized Radau points. All theoretical findings are confirmed by numerical experiments.

An hp-version Spectral Collocation Method for Nonlinear Volterra Integro-differential Equation with Weakly Singular Kernels

In this paper, we present an hp-version Legendre–Jacobi spectral collocation method for the nonlinear Volterra integro-differential equations with weakly singular kernels. We derive hp-version error bounds of the collocation method under the \(H^1\)-norm for the Volterra integro-differential equations with smooth solutions on arbitrary meshes and singular solutions on quasi-uniform meshes. Numerical experiments demonstrate the effectiveness of the proposed method.

A piecewise linear FEM for an optimal control problem of fractional operators: error analysis on curved domains

We propose and analyze a new discretization technique for a linear-quadratic optimal control problem involving the fractional powers of a symmetric and uniformly elliptic second order operator; control constraints are considered. Since these fractional operators can be realized as the Dirichlet-to-Neumann map for a nonuniformly elliptic equation, we recast our problem as a nonuniformly elliptic optimal control problem. The rapid decay of the solution to this problem suggests a truncation that is suitable for numerical approximation. We propose a fully discrete scheme that is based on piecewise linear functions on quasi-uniform meshes to approximate the optimal control and first-degree tensor product functions on anisotropic meshes for the optimal state variable. We provide an a priori error analysis that relies on derived Holder and Sobolev regularity estimates for the optimal variables and error estimates for a scheme that approximates fractional diffusion on curved domains; the latter being an extension of previous available results. The analysis is valid in any dimension. We conclude by presenting some numerical experiments that validate the derived error estimates.

Analyzing the Grell–Freitas Convection Scheme from Hydrostatic to Nonhydrostatic Scales within a Global Model

Abstract The authors implemented the Grell–Freitas (GF) parameterization of convection in which the cloud-base mass flux varies quadratically as a function of the convective updraft fraction in the global nonhydrostatic Model for Prediction Across Scales (MPAS). They evaluated the performance of GF using quasi-uniform meshes and a variable-resolution mesh centered over South America, the resolution of which varied between hydrostatic (50 km) and nonhydrostatic (3 km) scales. Four-day forecasts using a 50-km and a 15-km quasi-uniform mesh, initialized with GFS data for 0000 UTC 10 January 2014, reveal that MPAS overestimates precipitation in the tropics relative to the Tropical Rainfall Measuring Mission Multisatellite Precipitation Analysis data. Results of 4-day forecasts using the variable-resolution mesh reveal that over the refined region of the mesh, GF performs as a precipitating shallow convective scheme, whereas over the coarse region of the mesh, GF acts as a conventional deep convective scheme. As horizontal resolution increases and subgrid-scale motions become increasingly resolved, the contribution of convective and grid-scale precipitation to the total precipitation decreases and increases, respectively. Probability density distributions of precipitation highlight a smooth transition in the partitioning between convective and grid-scale precipitation, including at gray-zone scales across the transition region between the coarsest and finest regions of the global mesh. Variable-resolution meshes spanning between hydrostatic and nonhydrostatic scales are shown to be ideal tools to evaluate the horizontal scale dependence of parameterized convective and grid-scale moist processes.

A priori error analysis of the BEM with graded meshes for the electric field integral equation on polyhedral surfaces

The Galerkin boundary element discretisations of the electric field integral equation (EFIE) on Lipschitz polyhedral surfaces suffer slow convergence rates when the underlying surface meshes are quasi-uniform and shape-regular. This is due to singular behaviour of the solution to this problem in neighbourhoods of vertices and edges of the surface. Aiming to improve convergence rates of the Galerkin boundary element method (BEM) for the EFIE on a Lipschitz polyhedral closed surface Γ, we employ anisotropic meshes algebraically graded towards the edges of Γ. We prove that on sufficiently graded meshes the h-version of the BEM with the lowest-order Raviart–Thomas elements regains (up to a small order of ε>0) an optimal convergence rate (i.e., the rate of the h-BEM on quasi-uniform meshes for smooth solutions).

Discretization of the Poisson equation with non-smooth data and emphasis on non-convex domains

Several approaches are discussed how to understand the solution of the Dirichlet problem for the Poisson equation when the Dirichlet data are non-smooth such as if they are in L 2 only. For the method of transposition (sometimes called very weak formulation) three spaces for the test functions are considered, and a regularity result is proved. An approach of Berggren is recovered as the method of transposition with the second variant of test functions. A further concept is the regularization of the boundary data combined with the weak solution of the regularized problem. The effect of the regularization error is studied. The regularization approach is the simplest to discretize. The discretization error is estimated for a sequence of quasi-uniform meshes. Since this approach turns out to be equivalent to Berggren's discretization his error estimates are rendered more precisely. Numerical tests show that the error estimates are sharp, in particular that the order becomes arbitrarily small when the maximal interior angle of the domain tends to 2 π .© 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1433–1454, 2016

Block steepest descent method based on nonoverlapping domain decomposition, I

AbstractIn the present paper we propose a new iterative method for the numerical solution of system of linear algebraic equations with symmetric positive definite and positive semidefinite matrices arising from the approximation of diffusion equations by the mixed hybrid finite element method on triangular and tetrahedral meshes. The method is based on a combination of the block steepest descent idea and nonoverlapping domain decomposition algorithm. We give the formulation of the method and a simple convergence estimate in the case of regular shaped quasiuniform meshes.

FDTD Method for Piecewise Homogeneous Dielectric Media

In this paper, we consider a numerical solution of Maxwell’s curl equations for piecewise uniform dielectric medium by the example of a one-dimensional problem. For obtaining the second order accuracy, the electric field grid node is placed into the permittivity discontinuity point of the medium. If the dielectric permittivity is large, the problem becomes singularly perturbed and a contrast structure appears. We propose a piecewise quasi-uniform mesh which resolves all characteristic solution parts of the problem (regular part, boundary layer and transition zone placed between them) in detail. The features of the mesh are discussed.

Symmetric Coupling of LDG–FEM and DG–BEM

We analyze a discontinuous Galerkin FEM---BEM scheme for a second order elliptic transmission problem posed in the three-dimensional space. The symmetric variational formulation is discretized by nonconforming Raviart---Thomas finite elements on a general partition of the interior domain coupled with discontinuous boundary elements on an independent quasi-uniform mesh of the transmission interface. We prove (almost) quasi-optimal convergence of the method and confirm the theory by a numerical experiment. In addition we consider the case when continuous rather than discontinuous boundary elements are used.

Adaptive Crouzeix–Raviart boundary element method

For the nonconforming Crouzeix-Raviart boundary elements from (N. Heuer and F.-J. Sayas, Numer. Math. 112 (2009) 381-401), we develop and analyze a posteriori error estimators based on the hh/2 methodology. We discuss the optimal rate of convergence for uniform mesh refinement, and present a numerical experiment with singular data where our adaptive algorithm recovers the optimal rate while uniform mesh refinement is sub-optimal. We also discuss the case of reduced regularity by standard geometric singularities to conjecture that, in this situation, non-uniformly refined meshes are not superior to quasi-uniform meshes for Crouzeix-Raviart boundary elements. Mathematics Subject Classification. 65N30, 65N38, 65N50, 65R20.

Asymptotically exact a posteriori error estimators for first-order div least-squares methods in local and global [formula omitted] norm

A new asymptotically exact a posteriori error estimator is developed for first-order div least-squares (LS) finite element methods. Let (uh,σh) be LS approximate solution for (u,σ=−A∇u). Then, E=‖A−1/2(σh+A∇uh)‖0 is asymptotically exact a posteriori error estimator for ‖A1/2∇(u−uh)‖0 or ‖A−1/2(σ−σh)‖0 depending on the order of approximate spaces for σ and u. For E to be asymptotically exact for ‖A1/2∇(u−uh)‖0, we require higher order approximation property for σ, and vice versa. When both A∇u and σ are approximated in the same order of accuracy, the estimator becomes an equivalent error estimator for both errors. The underlying mesh is only required to be shape regular, i.e., it does not require quasi-uniform mesh nor any special structure for the underlying meshes. Confirming numerical results are provided and the performance of the estimator is explored for other choice of spaces for (uh,σh).

H-p Spectral element methods for three dimensional elliptic problems on non-smooth domains, Part-I: Regularity estimates and stability theorem

This is the first of a series of papers devoted to the study of h-p spectral element methods for solving three dimensional elliptic boundary value problems on non-smooth domains using parallel computers. In three dimensions there are three different types of singularities namely; the vertex, the edge and the vertex-edge singularities. In addition, the solution is anisotropic in the neighbourhoods of the edges and vertex-edges. To overcome the singularities which arise in the neighbourhoods of vertices, vertex-edges and edges, we use local systems of coordinates. These local coordinates are modified versions of spherical and cylindrical coordinate systems in their respective neighbourhoods. Away from these neighbourhoods standard Cartesian coordinates are used. In each of these neighbourhoods we use a geometrical mesh which becomes finer near the corners and edges. The geometrical mesh becomes a quasi-uniform mesh in the new system of coordinates. We then derive differentiability estimates in these new set of variables and state our main stability estimate theorem using a non-conforming h-p spectral element method whose proof is given in a separate paper.

An adaptive octree finite element method for PDEs posed on surfaces

Abstract The paper develops a finite element method for partial differential equations posed on hypersurfaces in R N , N = 2 , 3 . The method uses traces of bulk finite element functions on a surface embedded in a volumetric domain. The bulk finite element space is defined on an octree grid which is locally refined or coarsened depending on error indicators and estimated values of the surface curvatures. The cartesian structure of the bulk mesh leads to easy and efficient adaptation process, while the trace finite element method makes fitting the mesh to the surface unnecessary. The number of degrees of freedom involved in computations is consistent with the two-dimension nature of surface PDEs. No parametrization of the surface is required; it can be given implicitly by a level set function. In practice, a variant of the marching cubes method is used to recover the surface with the second order accuracy. We prove the optimal order of accuracy for the trace finite element method in H 1 and L 2 surface norms for a problem with smooth solution and quasi-uniform mesh refinement. Experiments with less regular problems demonstrate optimal convergence with respect to the number of degrees of freedom, if grid adaptation is based on an appropriate error indicator. The paper shows results of numerical experiments for a variety of geometries and problems, including advection–diffusion equations on surfaces. Analysis and numerical results of the paper suggest that combination of cartesian adaptive meshes and the unfitted (trace) finite elements provide simple, efficient, and reliable tool for numerical treatment of PDEs posed on surfaces.

Supercloseness of orthogonal projections onto nearby finite element spaces

We derive upper bounds on the difference between the orthogonal projections of a smooth function u onto two finite element spaces that are nearby, in the sense that the support of every shape function belonging to one but not both of the spaces is contained in a common region whose measure tends to zero under mesh refinement. The bounds apply, in particular, to the setting in which the two finite element spaces consist of continuous functions that are elementwise polynomials over shape-regular, quasi-uniform meshes that coincide except on a region of measure O(h γ ), where γ is a nonnegative scalar and h is the mesh spacing. The projector may be, for example, the orthogonal projector with respect to the L 2 -o rH 1 -inner product. In these and other circumstances, the bounds are superconvergent under a few mild regularity assumptions. That is, under mesh refinement, the two projections differ in norm by an amount that decays to zero at a faster rate than the amounts by which each projection differs from u. We present numerical examples to illustrate these superconvergent estimates and verify the necessity of the regularity assumptions on u.

Finite Elements with mesh refinement for wave equations in polygons

Error estimates for the space semi-discrete approximation of solutions of the Wave equation in polygons G⊂R2 are presented. Based on corner asymptotics of the solution, it is shown that for continuous, simplicial Lagrangian Finite Elements of polynomial degree p≥1 with either suitably graded mesh refinement or with bisection tree mesh refinement towards the corners of G, the maximal rate of convergence O(N−p/2) which is afforded by the Lagrangian Finite Element approximations on quasiuniform meshes for smooth solutions is restored. Dirichlet, Neumann and mixed boundary conditions are considered. Numerical experiments which confirm the theoretical results are presented. Generalizations to nonhomogeneous coefficients and elasticity and electromagnetics are indicated.

An $h$-$p$ Version of the Continuous Petrov--Galerkin Finite Element Method for Volterra Integro-Differential Equations with Smooth and Nonsmooth Kernels

We present an $h$-$p$ version of the continuous Petrov--Galerkin (CPG) finite element method for linear Volterra integro-differential equations with smooth and nonsmooth kernels. We establish a priori error estimates in the $L^2$-, $H^1$-, and $L^\infty$-norms that are completely explicit with respect to the local discretization and regularity parameters. For singular solutions caused by the weakly singular kernels, we prove that algebraic convergence rates can be achieved for the $h$-$p$ version of the CPG method with quasi-uniform meshes. Moreover, we show that exponential rates of convergence can be achieved for solutions with start-up singularities by using geometric time partitions and linearly increasing polynomial degrees. Numerical experiments are provided to illustrate the theoretical results.

Superconvergence for Neumann boundary control problems governed by semilinear elliptic equations

This paper is concerned with the discretization error analysis of semilinear Neumann boundary control problems in polygonal domains with pointwise inequality constraints on the control. The approximations of the control are piecewise constant functions. The state and adjoint state are discretized by piecewise linear finite elements. In a postprocessing step approximations of locally optimal controls of the continuous optimal control problem are constructed by the projection of the respective discrete adjoint state. Although the quality of the approximations is in general affected by corner singularities a convergence order of $h^2|\ln h|^{3/2}$ is proven for domains with interior angles smaller than $2\pi/3$ using quasi-uniform meshes. For larger interior angles mesh grading techniques are used to get the same order of convergence.