A-Posteriori Error Estimates for Discontinuous Galerkin Approximations of Second Order Elliptic Problems

Using the weighted residual formulation we derive a-posteriori estimates for Discontinuous Galerkin approximations of second order elliptic problems in mixed form. We show that our approach allows to include in a unified way all the methods presented so far in the literature.     

A Systematic Study on Weak Galerkin Finite Element Methods for Second Order Elliptic Problems

This article provides a systematic study for the weak Galerkin (WG) finite element method for second order elliptic problems by exploring polynomial approximations with various degrees for each local element. A typical local WG element is of the form \(P_k(T)\times P_j(\partial T)\Vert P_\ell (T)^2\), where \(k\ge 1\) is the degree of polynomials in the interior of the element T, \(j\ge 0\) is the degree of polynomials on the boundary of T, and \(\ell \ge 0\) is the degree of polynomials employed in the computation of weak gradients or weak first order partial derivatives. A general framework of stability and error estimate is developed for the corresponding numerical solutions. Numerical results are presented to confirm the theoretical results. The work reveals some previously undiscovered strengths of the WG method for second order elliptic problems, and the results are expected to be generalizable to other type of partial differential equations.     

A Posteriori Error Estimates for Weak Galerkin Finite Element Methods for Second Order Elliptic Problems

A residual type a posteriori error estimator is presented and analyzed for Weak Galerkin finite element methods for second order elliptic problems. The error estimator is proved to be efficient and reliable through two estimates, one from below and the other from above, in terms of an $H^1$ -equivalent norm for the exact error. Two numerical experiments are conducted to demonstrate the effectiveness of adaptive mesh refinement guided by this estimator.     

A Uniform Additive Schwarz Preconditioner for High-Order Discontinuous Galerkin Approximations of Elliptic Problems

In this paper we design and analyze a uniform preconditioner for a class of high-order Discontinuous Galerkin schemes. The preconditioner is based on a space splitting involving the high-order conforming subspace and results from the interpretation of the problem as a nearly-singular problem. We show that the proposed preconditioner exhibits spectral bounds that are uniform with respect to the discretization parameters, i.e., the mesh size, the polynomial degree and the penalization coefficient. The theoretical estimates obtained are supported by numerical tests.     

Direct Discontinuous Galerkin Method and Its Variations for Second Order Elliptic Equations

In this paper, we study direct discontinuous Galerkin method (Liu and Yan in SIAM J Numer Anal 47(1):475–698, 2009) and its variations (Liu and Yan in Commun Comput Phys 8(3):541–564, 2010; Vidden and Yan in J Comput Math 31(6):638–662, 2013; Yan in J Sci Comput 54(2–3):663–683, 2013) for 2nd order elliptic problems. A priori error estimate under energy norm is established for all four methods. Optimal error estimate under \(L^2\) norm is obtained for DDG method with interface correction (Liu and Yan in Commun Comput Phys 8(3):541–564, 2010) and symmetric DDG method (Vidden and Yan in J Comput Math 31(6):638–662, 2013). A series of numerical examples are carried out to illustrate the accuracy and capability of the schemes. Numerically we obtain optimal \((k+1)\)th order convergence for DDG method with interface correction and symmetric DDG method on nonuniform and unstructured triangular meshes. An interface problem with discontinuous diffusion coefficients is investigated and optimal \((k+1)\)th order accuracy is obtained. Peak solutions with sharp transitions are captured well. Highly oscillatory wave solutions of Helmholz equation are well resolved.     

Perfectly Matched Layers for Time-Harmonic Second Order Elliptic Problems

The main goal of this work is to give a review of the Perfectly Matched Layer (PML) technique for time-harmonic problems. Precisely, we focus our attention on problems stated in unbounded domains, which involve second order elliptic equations writing in divergence form and, in particular, on the Helmholtz equation at low frequency regime. Firstly, the PML technique is introduced by means of a simple porous model in one dimension. It is emphasized that an adequate choice of the so called complex absorbing function in the PML yields to accurate numerical results. Then, in the two-dimensional case, the PML governing equation is described for second order partial differential equations by using a smooth complex change of variables. Its mathematical analysis and some particular examples are also included. Numerical drawbacks and optimal choice of the PML absorbing function are studied in detail. In fact, theoretical and numerical analysis show the advantages of using non-integrable absorbing functions. Finally, we present some relevant real life numerical simulations where the PML technique is widely and successfully used although they are not covered by the standard theoretical framework.     

Interior Penalty Discontinuous Galerkin Methods for Second Order Linear Non-divergence Form Elliptic PDEs

This paper develops interior penalty discontinuous Galerkin (IP-DG) methods to approximate \(W^{2,p}\) strong solutions of second order linear elliptic partial differential equations (PDEs) in non-divergence form with continuous coefficients. The proposed IP-DG methods are closely related to the IP-DG methods for advection-diffusion equations, and they are easy to implement on existing standard IP-DG software platforms. It is proved that the proposed IP-DG methods have unique solutions and converge with optimal rate to the \(W^{2,p}\) strong solution in a discrete \(W^{2,p}\)-norm. The crux of the analysis is to establish a DG discrete counterpart of the Calderon–Zygmund estimate and to adapt a freezing coefficient technique used for the PDE analysis at the discrete level. To obtain such a crucial estimate, we need to establish broken \(W^{1,p}\)-norm error estimates for IP-DG approximations of constant coefficient elliptic PDEs, which is also of independent interest. Numerical experiments are provided to gauge the performance of the proposed IP-DG methods and to validate the theoretical convergence results.     

Local Discontinuous Galerkin Methods for One-Dimensional Second Order Fully Nonlinear Elliptic and Parabolic Equations

This paper is concerned with developing accurate and efficient nonstandard discontinuous Galerkin methods for fully nonlinear second order elliptic and parabolic partial differential equations (PDEs) in the case of one spatial dimension. The primary goal of the paper to develop a general framework for constructing high order local discontinuous Galerkin (LDG) methods for approximating viscosity solutions of these fully nonlinear PDEs which are merely continuous functions by definition. In order to capture discontinuities of the first order derivative $u_x$ of the solution $u$ , two independent functions $q^-$ and $q^+$ are introduced to approximate one-sided derivatives of $u$ . Similarly, to capture the discontinuities of the second order derivative $u_{xx}$ , four independent functions $p^{- -}, p^{- +}, p^{+ -}$ , and $p^{+ +}$ are used to approximate one-sided derivatives of $q^-$ and $q^+$ . The proposed LDG framework, which is based on a nonstandard mixed formulation of the underlying PDE, embeds a given fully nonlinear problem into a mostly linear system of equations where the given nonlinear differential operator must be replaced by a numerical operator which allows multiple value inputs of the first and second order derivatives $u_x$ and $u_{xx}$ . An easy to verify set of criteria for constructing “good” numerical operators is also proposed. It consists of consistency and generalized monotonicity. To ensure such a generalized monotonicity property, the crux of the construction is to introduce the numerical moment in the numerical operator, which plays a critical role in the proposed LDG framework. The generalized monotonicity gives the LDG methods the ability to select the viscosity solution among all possible solutions. The proposed framework extends a companion finite difference framework developed by Feng and Lewis (J Comp Appl Math 254:81–98, 2013) and allows for the approximation of fully nonlinear PDEs using high order polynomials and non-uniform meshes. Numerical experiments are also presented to demonstrate the accuracy, efficiency and utility of the proposed LDG methods.     

Nonstandard Local Discontinuous Galerkin Methods for Fully Nonlinear Second Order Elliptic and Parabolic Equations in High Dimensions

This paper is concerned with developing accurate and efficient numerical methods for fully nonlinear second order elliptic and parabolic partial differential equations (PDEs) in multiple spatial dimensions. It presents a general framework for constructing high order local discontinuous Galerkin (LDG) methods for approximating viscosity solutions of these fully nonlinear PDEs. The proposed LDG methods are natural extensions of a narrow-stencil finite difference framework recently proposed by the authors for approximating viscosity solutions. The idea of the methodology is to use multiple approximations of first and second order derivatives as a way to resolve the potential low regularity of the underlying viscosity solution. Consistency and generalized monotonicity properties are proposed that ensure the numerical operator approximates the differential operator. The resulting algebraic system has several linear equations coupled with only one nonlinear equation that is monotone in many of its arguments. The structure can be explored to design nonlinear solvers. This paper also presents and analyzes numerical results for several numerical test problems in two dimensions which are used to gauge the accuracy and efficiency of the proposed LDG methods.     

Superconvergent Discontinuous Galerkin Methods for Linear Non-selfadjoint and Indefinite Elliptic Problems

Based on Cockburn et al. (Math. Comp. 78:1?C24, 2009), superconvergent discontinuous Galerkin methods are identified for linear non-selfadjoint and indefinite elliptic problems. With the help of an auxiliary problem which is the discrete version of a linear non-selfadjoint elliptic problem in divergence form, optimal error estimates of order k+1 in L ²-norm for the potential and the flux are derived, when piecewise polynomials of degree k??1 are used to approximate both potential and flux variables. Using a suitable post-processing of the discrete potential, it is then shown that the resulting post-processed potential converges with order k+2 in L ²-norm. The article is concluded with a numerical experiment which confirms the theoretical results.     

Error Boundedness of Discontinuous Galerkin Spectral Element Approximations of Hyperbolic Problems

We examine the long time error behavior of discontinuous Galerkin spectral element approximations to hyperbolic equations. We show that the choice of numerical flux at interior element boundaries affects the growth rate and asymptotic value of the error. Using the upwind flux, the error reaches the asymptotic value faster, and to a lower value than a central flux gives, especially for low resolution computations. The differences in the error caused by the numerical flux choice decrease as the solution becomes better resolved.     

Adaptive Multilevel Correction Method for Finite Element Approximations of Elliptic Optimal Control Problems

In this paper we propose an adaptive multilevel correction scheme to solve optimal control problems discretized with finite element method. Different from the classical adaptive finite element method (AFEM for short) applied to optimal control which requires the solution of the optimization problem on new finite element space after each mesh refinement, with our approach we only need to solve two linear boundary value problems on current refined mesh and an optimization problem on a very low dimensional space. The linear boundary value problems can be solved with well-established multigrid method designed for elliptic equation and the optimization problems are of small scale corresponding to the space built with the coarsest space plus two enriched bases. Our approach can achieve the similar accuracy with standard AFEM but greatly reduces the computational cost. Numerical experiments demonstrate the efficiency of our proposed algorithm.     

Domain Decomposition Preconditioners for Discontinuous Galerkin Methods for Elliptic Problems on Complicated Domains

In this article we consider the application of Schwarz-type domain decomposition preconditioners for discontinuous Galerkin finite element approximations of elliptic partial differential equations posed on complicated domains, which are characterized by small details in the computational domain or microstructures. In this setting, it is necessary to define a suitable coarse-level solver, in order to guarantee the scalability of the preconditioner under mesh refinement. To this end, we exploit recent ideas developed in the so-called composite finite element framework, which allows for the definition of finite element methods on general meshes consisting of agglomerated elements. Numerical experiments highlighting the practical performance of the proposed preconditioner are presented.     

A Posteriori Error Estimates of Discontinuous Galerkin Method for Nonmonotone Quasi-linear Elliptic Problems

In this paper, we propose and study the residual-based a posteriori error estimates of h-version of symmetric interior penalty discontinuous Galerkin method for solving a class of second order quasi-linear elliptic problems which are of nonmonotone type. Computable upper and lower bounds on the error measured in terms of a natural mesh-dependent energy norm and the broken H ¹-seminorm, respectively, are derived. Numerical experiments are also provided to illustrate the performance of the proposed estimators.     

Exponentially Accurate Spectral Element Method for Fourth Order Elliptic Problems

In this paper, a fully non-conforming least-squares spectral element method for fourth order elliptic problems on smooth domains is presented. The proposed method works for a general fourth order elliptic operator with non-homogeneous data in two dimensions and gives exponentially accurate solutions. We derive differentiability estimates and prove our main stability estimate theorem using a non-conforming spectral element method. We then formulate a numerical scheme using a block diagonal preconditioner. Error estimates are also proven for the proposed method. We provide the computational complexity of our method and present results of numerical simulations that have been performed to validate the theory.     

A Bivariate Spline Method for Second Order Elliptic Equations in Non-divergence Form

A bivariate spline method is developed to numerically solve second order elliptic partial differential equations in non-divergence form. The existence, uniqueness, stability as well as approximation properties of the discretized solution will be established by using the well-known Ladyzhenskaya–Babuska–Brezzi condition. Bivariate splines, discontinuous splines with smoothness constraints are used to implement the method. Computational results based on splines of various degrees are presented to demonstrate the effectiveness and efficiency of our method.     

Second Order Methods for the Optimal Control Problems

The paper considers the iterative improvement algorithms, the efficiency of which substantially depends on the chosen parameters values. The problem of control of these parameters is formulated and discussed. We designed the modified algorithms where parameters are automatically adjusted at each iteration. The original and modified algorithms are applied to solve the problem of optimal control for the ecology-economic system.     

Multi-scale Discontinuous Galerkin Method for Solving Elliptic Problems with Curvilinear Unidirectional Rough Coefficients

In this paper, we propose a multi-scale discontinuous Galerkin (DG) method for second-order elliptic problems with curvilinear unidirectional rough coefficients by choosing a special non-polynomial approximation space. The key ingredient of the method lies in the incorporation of the local oscillatory features of the differential operators into the approximation space so as to capture the multi-scale solutions without having to resolve the finest scales. The unidirectional feature of the rough coefficients allows us to construct the basis functions of the DG non-polynomial approximation space explicitly, thereby greatly increasing the algorithm efficiency. Detailed error estimates for two-dimensional second-order DG methods are derived, and a general guidance on how to construct such non-polynomial basis is discussed. Numerical examples are also presented to validate and demonstrate the effectiveness of the algorithm.     

A Weak Galerkin Method with an Over-Relaxed Stabilization for Low Regularity Elliptic Problems

A new weak Galerkin (WG) finite element method is developed and analyzed for solving second order elliptic problems with low regularity solutions in the Sobolev space \(W^{2,p}(\Omega )\) with \(p\in (1,2)\). A WG stabilizer was introduced by Wang and Ye (Math Comput 83:2101–2126, 2014) for a simpler variational formulation, and it has been commonly used since then in the WG literature. In this work, for the purpose of dealing with low regularity solutions, we propose to generalize the stabilizer of Wang and Ye by introducing a positive relaxation index to the mesh size h. The relaxed stabilization gives rise to a considerable flexibility in treating weak continuity along the interior element edges. When the norm index \(p\in (1,2]\), we strictly derive that the WG error in energy norm has an optimal convergence order \(O(h^{l+1-\frac{1}{p}-\frac{p}{4}})\) by taking the relaxed factor \(\beta =1+\frac{2}{p}-\frac{p}{2}\), and it also has an optimal convergence order \(O(h^{l+2-\frac{2}{p}})\) in \(L^2\) norm when the solution \(u\in W^{l+1,p}\) with \(p\in [1,1+\frac{2}{p}-\frac{p}{2}]\) and \(l\ge 1\). It is recovered for \(p=2\) that with the choice of \(\beta =1\), error estimates in the energy and \(L^2\) norms are optimal for the source term in the sobolev space \(L^2\). Weak variational forms of the WG method give rise to desirable flexibility in enforcing boundary conditions and can be easily implemented without requiring a sufficiently large penalty factor as in the usual discontinuous Galerkin methods. In addition, numerical results illustrate that the proposed WG method with an over-relaxed factor \(\beta (\ge 1)\) converges at optimal algebraic rates for several low regularity elliptic problems.