In this work we analyze the inverse problem of recovering the space-dependent potential coefficient in an elliptic / parabolic problem from distributed observation. We establish novel (weighted) conditional stability estimates under very mild conditions on the problem data. Then we provide an error analysis of a standard reconstruction scheme based on the standard output least-squares formulation with Tikhonov regularization (by an $H^1$-seminorm penalty), which is then discretized by the Galerkin finite element method with continuous piecewise linear finite elements in space (and also backward Euler method in time for parabolic problems). We present a detailed analysis of the discrete scheme, and provide convergence rates in a weighted $L^2(Ω)$ for discrete approximations with respect to the exact potential. The error bounds are explicitly dependent on the noise level, regularization parameter and discretization parameter(s). Under suitable conditions, we also derive error estimates in the standard $L^2(Ω)$ and interior $L^2$ norms. The analysis employs sharp a priori error estimates and nonstandard test functions. Several numerical experiments are given to complement the theoretical analysis.
A discontinuous Galerkin pressure correction numerical method for solving the incompressible Navier-Stokes equations is formulated and analyzed. We prove unconditional stability of the propose scheme. Convergence of the discrete velocity is established by deriving a priori error estimates. Numerical results verify the convergence rates.
The hybrid Raviart-Thomas mixed discontinuous Galerkin (HRTMDG) method is proposed for solving the Helmholtz equation. With a new energy norm, we establish the existence and uniqueness of the HRTMDG method, and give its convergence analysis. The corresponding error estimate shows that the HRTMDG method has an optimal $L^2$-norm convergence accuracy which is independent of wavenumber.
In the present paper, we prove convergence rates for the Local Discontinuous Galerkin (LDG) approximation, proposed in Part I of the paper, for systems of $p$-Navier-Stokes type and $p$-Stokes type with $p\in (2,\infty)$. The convergence rates are optimal for linear ansatz functions. The results are supported by numerical experiments.
In this paper, we investigate a Local Discontinuous Galerkin (LDG) approximation for systems with balanced Orlicz-structure. We propose a new numerical flux, which yields optimal convergence rates for linear ansatz functions. In particular, our approach yields a unified treatment for problems with $(p,δ)$-structure for arbitrary $p \in (1,\infty)$ and $δ\ge 0$.
In this paper, we present convergence analysis of high-order finite element based methods, in particular, we focus on a discontinuous Galerkin scheme using summation-by-parts operators. To this end, it is crucial that structure preserving properties, such as positivity preservation and entropy inequality hold. We demonstrate how to ensure them and prove the convergence of our multidimensional high-order DG scheme via dissipative weak solutions. In numerical simulations, we verify our theoretical results.
We prove the convergence of discontinuous Galerkin approximations for the Vlasov-Poisson system written as an hyperbolic system using Hermite polynomials in velocity. To obtain stability properties, we introduce a suitable weighted L 2 space, with a time dependent weight, and first prove global stability for the weighted L 2 norm and propagation of regularity. Then we prove error estimates between the numerical solution and the smooth solution to the Vlasov-Poisson system.
In the present paper, we prove convergence rates for the pressure of the Local Discontinuous Galerkin (LDG) approximation, proposed in Part I of the paper, of systems of $p$-Navier-Stokes type and $p$-Stokes type with $p\in (2,\infty)$. The results are supported by numerical experiments.
In this paper, we study the convergence of adaptive mixed interior penalty discontinuous Galerkin method for H(curl)-elliptic problems. We first get the mixed model of H(curl)-elliptic problem by introducing a new intermediate variable. Then we discuss the continuous variational problem and discrete variational problem, which based on interior penalty discontinuous Galerkin approximation. Next, we construct the corresponding posteriori error indicator, and prove the contraction of the summation of the energy error and the scaled error indicator. At last, we confirm and illustrate the theoretic…
This paper presents and analyzes a discontinuous Galerkin method for the incompressible three-phase flow problem in porous media. We use a first order time extrapolation which allows us to solve the equations implicitly and sequentially. We show that the discrete problem is well-posed, and obtain a priori error estimates. Our numerical results validate the theoretical results, i.e. the algorithm converges with first order.
In the present paper, we propose a Local Discontinuous Galerkin (LDG) approximation for fully non-homogeneous systems of $p$-Navier-Stokes type. On the basis of the primal formulation, we prove well-posedness, stability (a priori estimates), and weak convergence of the method. To this end, we propose a new DG discretization of the convective term and develop an abstract non-conforming theory of pseudo-monotonicity, which is applied to our problem. We also use our approach to treat the $p$-Stokes problem.
In this paper, we theoretically and numerically verify that the discontinuous Galerkin (DG) methods with central fluxes for linear hyperbolic equations on non-uniform meshes have sub-optimal convergence properties when measured in the $L^2$-norm for even degree polynomial approximations. On uniform meshes, the optimal error estimates are provided for arbitrary number of cells in one and multi-dimensions, improving previous results. The theoretical findings are found to be sharp and consistent with numerical results.
The subject of this work is a new stochastic Galerkin method for second-order elliptic partial differential equations with random diffusion coefficients. It combines operator compression in the stochastic variables with tree-based spline wavelet approximation in the spatial variables. Relying on a multilevel expansion of the given random diffusion coefficient, the method is shown to achieve optimal computational complexity up to a logarithmic factor. In contrast to existing results, this holds in particular when the achievable convergence rate is limited by the regularity of the random field, …
We develop a general convergence theory for adaptive discontinuous Galerkin methods for elliptic PDEs covering the popular SIPG, NIPG and LDG schemes as well as all practically relevant marking strategies. Another key feature of the presented result is, that it holds for penalty parameters only necessary for the standard analysis of the respective scheme. The analysis is based on a quasi interpolation into a newly developed limit space of the adaptively created non-conforming discrete spaces, which enables to generalise the basic convergence result for conforming adaptive finite element method…
We develop a basic convergence analysis for an adaptive $\textsf{C}^0\textsf{IPG}$ method for the Biharmonic problem, which provides convergence without rates for all practically relevant marking strategies and all penalty parameters assuring coercivity of the method. The analysis hinges on embedding properties of (broken) Sobolev and BV spaces, and the construction of a suitable limit space. In contrast to the convergence result of adaptive discontinuous Galerkin methods for elliptic PDEs, by Kreuzer and Georgoulis [Math. Comp. 87 (2018), no.~314, 2611--2640], here we have to deal with the fa…
We present an error analysis for the discontinuous Galerkin method applied to the discrete-ordinate discretization of the steady-state radiative transfer equation. Under some mild assumptions, we show that the DG method converges uniformly with respect to a scaling parameter $\varepsilon$ which characterizes the strength of scattering in the system. However, the rate is not optimal and can be polluted by the presence of boundary layers. In one-dimensional slab geometries, we demonstrate optimal convergence when boundary layers are not present and analyze a simple strategy for balance interior …
Existing a priori convergence results of the discontinuous Petrov-Galerkin method to solve the problem of linear elasticity are improved. Using duality arguments, we show that higher convergence rates for the displacement can be obtained. Post-processing techniques are introduced in order to prove superconvergence and numerical experiments {\color{black} confirm} our theory.
For singularly perturbed reaction-diffusion problems in 1D and 2D, we study a local discontinuous Galerkin (LDG) method on a Shishkin mesh. In these cases, the standard energy norm is too weak to capture adequately the behavior of the boundary layers that appear in the solutions. To deal with this deficiency, we introduce a balanced norm stronger than the energy norm. In order to achieve optimal convergence under the balanced norm in one-dimensional case, we design novel numerical fluxes and propose a special interpolation that consists of a Gauss-Radau projection and a local $L^2$ projection.…
In this paper, we present a study on the effect of penalty term in the mixed Discontinuous Galerkin Finite Element Method for the biharmonic equation proposed by \cite{gudi2008mixed}. The proposed mixed Discontinuous Galerkin Method showed sub-optimal rates of convergence for piecewise quadratic elements and no significant convergence rates for piecewise linear elements. We show that by choosing the penalty term proportional to $|e_k|^{-1}$ instead of $|e_k|^{-3}$, ensures an optimal rate of convergence for the approximation, including for piecewise linear elements. Finally, we present numeric…
In this paper, we provide a framework of designing the local discontinuous Galerkin scheme for integral fractional Laplacian $(-Δ)^{s}$ with $s\in(0,1)$ in two dimensions. We theoretically prove and numerically verify the numerical stability and convergence of the scheme with the convergence rate no worse than $\mathcal{O}(h^{k+\frac{1}{2}})$.
We generalize the energy-based discontinuous Galerkin method proposed in [SIAM J. Num. Anal., 53(6):2705-2726, 2015.] to second-order semilinear wave equations. A stability and convergence analysis is presented along with numerical experiments demonstrating optimal convergence for certain choices of the interelement fluxes. Applications to the sine-Gordon equation include simulations of breathers, kink, and anti-kink solitons.
We consider spectral mixed discontinuous Galerkin finite element discretizations of the Lamé system of linear elasticity in polyhedral domains in $\mathbb{R}^3$. In order to resolve possible corner, edge, and corner-edge singularities, anisotropic geometric edge meshes consisting of hexahedral elements are applied. We perform a computational study on the discrete inf-sup stability of these methods, and especially focus on the robustness with respect to the Poisson ratio close to the incompressible limit (i.e. the Stokes system). Furthermore, under certain realistic assumptions (for analytic da…
Stochastic Galerkin Method for Fractional Boundary Value Problems: Convergence Analysis and Numerical Treatment
Indian Institute of Technology (Banaras Hindu University)（印度理工大学（贝拿勒斯印度教大学））；Birla Institute of Technology and Science, Pilani（比拉理工学院，皮拉尼校区）；University of Greifswald（格赖夫斯瓦尔德大学）
Standard low-order finite elements, which perform well for problems involving compressible elastic materials, are known to under-perform when nearly incompressible materials are involved, commonly exhibiting the locking phenomenon. Interior penalty (IP) discontinuous Galerkin methods have been shown to circumvent locking when simplicial elements are used. The same IP methods, however, result in locking on meshes of quadrilaterals. The authors have shown in earlier work that under-integration of specified terms in the IP formulation eliminates the locking problem for rectangular elements. Here …
This paper analyzes a time-stepping discontinuous Galerkin method for fractional diffusion-wave problems. This method uses piecewise constant functions in the temporal discretization and continuous piecewise linear functions in the spatial discretization. Nearly optimal convergence rate with respect to the regularity of the solution is established when the source term is nonsmooth, and nearly optimal convergence rate $ \ln(1/τ)(\sqrt{\ln(1/h)} h^2+τ)$ is derived under appropriate regularity assumption on the source term. Convergence is also established without smoothness assumption on the init…
In this paper, we develop a convergence theory for Dual-Primal Isogeometric Tearing and Interconnecting (IETI-DP) solvers for isogeometric multi-patch discretizations of the Poisson problem, where the patches are coupled using discontinuous Galerkin. The presented theory provides condition number bounds that are explicit in the grid sizes h and in the spline degrees p. We give an analysis that holds for various choices for the primal degrees of freedom: vertex values, edge averages, and a combination of both. If only the vertex values or both vertex values and edge averages are taken as primal…
Fourier continuation is an approach used to create periodic extensions of non-periodic functions in order to obtain highly-accurate Fourier expansions. These methods have been used in PDE-solvers and have demonstrated high-order convergence and spectrally accurate dispersion relations in numerical experiments. Discontinuous Galerkin (DG) methods are increasingly used for solving PDEs and, as all Galerkin formulations, come with a strong framework for proving stability and convergence. Here we propose the use of Fourier continuation in forming a new basis for the DG framework.
We address the issue of the suboptimality in the p-version discontinuous Galerkin (dG) methods for first order hyperbolic problems. The convergence rate is derived for the upwind dG scheme on tensor product meshes in any dimension. The standard proof in seminal work [14] leads to suboptimal convergence in terms of the polynomial degree by 3/2 order for general convection fields, with the exception of piecewise multi-linear convection fields, which rather yield optimal convergence. Such suboptimality is not observed numerically. Thus, it might be caused by a limitation of the analysis, which we…
We present a Reduced Order Model (ROM) which exploits recent developments in Physics Informed Neural Networks (PINNs) for solving inverse problems for the Navier--Stokes equations (NSE). In the proposed approach, the presence of simulated data for the fluid dynamics fields is assumed. A POD-Galerkin ROM is then constructed by applying POD on the snapshots matrices of the fluid fields and performing a Galerkin projection of the NSE (or the modified equations in case of turbulence modeling) onto the POD reduced basis. A $\textit{POD-Galerkin PINN ROM}$ is then derived by introducing deep neural …
This paper generalizes the earlier work on the energy-based discontinuous Galerkin method for second-order wave equations to fourth-order semilinear wave equations. We first rewrite the problem into a system with a second-order spatial derivative, then apply the energy-based discontinuous Galerkin method to the system. The proposed scheme, on the one hand, is more computationally efficient compared with the local discontinuous Galerkin method because of fewer auxiliary variables. On the other hand, it is unconditionally stable without adding any penalty terms, and admits optimal convergence in…
This work is devoted to an inverse problem of identifying a source term depending on both spatial and time variables in a parabolic equation from single Cauchy data on a part of the boundary. A Crank-Nicolson Galerkin method is applied to the least squares functional with an quadratic stabilizing penalty term. The convergence of finite dimensional regularized approximations to the sought source as measurement noise levels and mesh sizes approach to zero with an appropriate regularization parameter is proved. Moreover, under a suitable source condition, an error bound and corresponding converge…
In this paper we discuss the convergence rate for Galerkin approximation of the stochastic Allen-Cahn equations driven by space-time white noise on $\T$. First we prove that the convergence rate for stochastic 2D heat equation is of order $α-δ$ in Besov space $\C^{-α}$ for $α\in(0,1)$ and $δ>0$ arbitrarily small. Then we obtain the convergence rate for Galerkin approximation of the stochastic Allen-Cahn equations of order $α-δ$ in $\C^{-α}$ for $α\in(0,2/9)$ and $δ>0$ arbitrarily small.
We study the numerical approximation by space-time finite element methods of a multi-physics system coupling hyperbolic elastodynamics with parabolic transport and modeling poro- and thermoelasticity. The equations are rewritten as a first-order system in time. Discretizations by continuous Galerkin methods in time and inf-sup stable pairs of finite element spaces for the spatial variables are investigated. Optimal order error estimates are proved by an analysis in weighted norms that depict the energy of the system's unknowns. A further important ingredient and challenge of the analysis is th…
An hp-version interior penalty discontinuous Galerkin (IPDG) method under nonconforming meshes is proposed to solve the quad-curl eigenvalue problem. We prove well-posedness of the numerical scheme for the quad-curl equation and then derive an error estimate in a mesh-dependent norm, which is optimal with respect to h but has different p-version error bounds under conforming and nonconforming tetrahedron meshes. The hp-version discrete compactness of the DG space is established for the convergence proof. The performance of the method is demonstrated by numerical experiments using conforming/no…
