Analysis of a continuous Galerkin method with mesh modification for two-dimensional telegraph equation

The study of a continuous Galerkin method for Sobolev equation with space-time variable coefficients

In Chs. 5 and 8 we discussed the implementation of continuous Galerkin (CG) methods in one dimension for hyperbolic and elliptic equations, respectively. In Ch. 12 we introduced the extension of the CG method to two dimensions by describing its implementation for elliptic partial differential equations (PDEs). In this chapter we extend the CG method for the application of hyperbolic equations in two dimensions. We also discuss the addition of diffusion operators.

We propose a new technique for computing highly accurate approximations to linear functionals in terms of Galerkin approximations. We illustrate the technique on a simple model problem, namely, that of the approximation of J(u), where $$J(\cdot )$$ is a very smooth functional and u is the solution of a Poisson problem; we assume that the solution u and the solution of the adjoint problem are both very smooth. It is known that, if $$u_h$$ is the approximation given by the continuous Galerkin method with piecewise polynomials of degree $$k>0$$ , then, as a direct consequence of its property of Galerkin orthogonality, the functional $$J(u_h)$$ converges to J(u) with a rate of order $$h^{2k}$$ . We show how to define approximations to J(u), with a computational effort about twice of that of computing $$J(u_h)$$ , which converge with a rate of order $$h^{4k}$$ . The new technique combines the adjoint-recovery method for providing precise approximate functionals by Pierce and Giles (SIAM Rev 42(2):247–264, 2000), which was devised specifically for numerical approximations without a Galerkin orthogonality property, and the accuracy-enhancing convolution technique of Bramble and Schatz (Math Comput 31(137):94–111, 1977), which was devised specifically for numerical methods satisfying a Galerkin orthogonality property, that is, for finite element methods like, for example, continuous Galerkin, mixed, discontinuous Galerkin and the so-called hybridizable discontinuous Galerkin methods. For the latter methods, we present numerical experiments, for $$k=1,2,3$$ in one-space dimension and for $$k=1,2$$ in two-space dimensions, which show that $$J(u_h)$$ converges to J(u) with order $$h^{2k+1}$$ and that the new approximations converges with order $$h^{4k}$$ . The numerical experiments also indicate, for the p-version of the method, that the rate of exponential convergence of the new approximations is about twice that of $$J(u_h)$$ .

Discontinuous and coupled continuous/discontinuous Galerkin methods for the shallow water equations

Fluorescence diffuse optical tomography is a powerful tool for the investigation of molecular events in studies for new therapeutic developments. Here, the emphasis is put on the mathematical problem of tomography, which can be formulated in terms of an estimation of physical parameters appearing as a set of Partial Differential Equations (PDEs). The standard polynomial Finite Element Method (FEM) is a method of choice to solve the diffusion equation because it has no restriction in terms of neither the geometry nor the homogeneity of the system, but it is time consuming. In order to speed up computation time, this paper proposes an alternative numerical model, describing the diffusion operator in orthonormal basis of compactly supported wavelets. The discretization of the PDEs yields to matrices which are easily computed from derivative wavelet product integrals. Due to the shape of the wavelet basis, the studied domain is included in a regular fictitious domain. A validation study and a comparison with the standard FEM are conducted on synthetic data.

A new space–time continuous Galerkin method with mesh modification for Sobolev equations

This dissertation, framed within an Industrial Doctorate programme, has the twofold objective of improving the understanding of numerical properties and convergence acceleration techniques for High-Order methods, and to formulate an efficient incompressible flow solver for thermal applications. The first part of this manuscript is devoted to the study of the linear multi-grid method applied to Flux Reconstruction, a relatively recent High-Order numerical method with close relations to the well-known Discontinuous Galerkin and Spectral Difference schemes. We do so by investigating the non-modal dissipation behaviour of the multi-grid scheme as a function of its parameters, in an effort to find the most efficient configurations. The analysis is extended to 2D high aspect-ratio cells, in the presence of dominating convection. The second part of this manuscript introduces a High-Order incompressible flow solver devised by the simultaneous use of the Flux Reconstruction, the Discontinuous Galerkin and the Continuous Galerkin methods, each of which is known to offer increased computational efficiency for certain types of problems. On the one end of the spectrum, the Flux Reconstruction is an efficient algorithm for the explicit evaluation of fluxes in conservative form; on the other end, the Continuous Galerkin method is comparatively very efficient for the resolution of diffusion-dominated steady problems. The different formulations are put together in a fractional-step approach and validated against known test cases – unsteady Stokes and Taylor problems, lid-driven cavity, Taylor-Green vortex and thermally-driven cavity – with and without analytical solution, demonstrating convergence with mesh, time-step and polynomial refinement. RESUMEN El presente trabajo, enmarcado en un programa de Doctorado Industrial, tiene el doble objetivo de mejorar la comprensión de las propiedades numéricas y técnicas de aceleración de convergencia para métodos de alto orden, así como de formular un solver eficiente para la resolución de flujos incompresibles con aplicaciones térmicas. La primera parte del manuscrito está dedicada al estudio del método multi-grid lineal aplicado a Flux Reconstruction, un método numérico de alto orden relativamente reciente y estrechamente relacionado con los conocidos Discontinuous Galerkin y Spectral Difference. Ello se consigue investigando la disipación numérica no-modal del esquema multi-grid en función de sus parámetros, con el objetivo de determinar aquellas configuraciones más eficientes. El análisis se extiende a 2D con elementos con alta relación de aspecto en presencia de convección dominante. La segunda parte del manuscrito presenta un solver de alto orden incompresible concebido con el uso simultáneo de los métodos Flux Reconstruction, Discontinuous Galerkin y Continuous Galerkin, cada uno de los cuales ofrece una eficiencia computacional superior para ciertos tipos de problemas. En un extremo, Flux Reconstruction es un algoritmo eficiente para la evaluación explícita de flujos en forma conservativa; en el otro extremo, Continuous Galerkin es comparativamente muy eficiente para la resolución de problemas elípticos en estado estacionario. Las diferentes formulaciones se conjugan en un método de paso fraccionado validado con conocidos casos test con y sin solución analítica, demostrando convergencia con el refinamiento en el mallado, el orden polinomial y el paso de tiempo.

The optimal rate of convergence of the wave equation in both the energy and theL2-norms using continuous Galerkin method is well known. We exploit this technique and design a fully discrete scheme consisting of coupling the nonstandard finite difference method in the time and the continuous Galerkin method in the space variables. We show that, for sufficiently smooth solution, the maximal error in theL2-norm possesses the optimal rate of convergenceO(h2+(Δt)2)wherehis the mesh size andΔtis the time step size. Furthermore, we show that this scheme replicates the properties of the exact solution of the wave equation. Some numerical experiments should be performed to support our theoretical analysis.

Mass conservation of the unified continuous and discontinuous element-based Galerkin methods on dynamically adaptive grids with application to atmospheric simulations

In the hyperbolic community, discontinuous Galerkin (DG) approaches are mainly applied when finite element methods are considered. As the name suggested, the DG framework allows a discontinuity at the element interfaces, which seems for many researchers a favorable property in case of hyperbolic balance laws. On the contrary, continuous Galerkin methods appear to be unsuitable for hyperbolic problems and there exists still the perception that continuous Galerkin methods are notoriously unstable. To remedy this issue, stabilization terms are usually added and various formulations can be found in the literature. However, this perception is not true and the stabilization terms are unnecessary, in general. In this paper, we deal with this problem, but present a different approach. We use the boundary conditions to stabilize the scheme following a procedure that are frequently used in the finite difference community. Here, the main idea is to impose the boundary conditions weakly and specific boundary operators are constructed such that they guarantee stability. This approach has already been used in the discontinuous Galerkin framework, but here we apply it with a continuous Galerkin scheme. No internal dissipation is needed even if unstructured grids are used. Further, we point out that we do not need exact integration, it suffices if the quadrature rule and the norm in the differential operator are the same, such that the summation-by-parts property is fulfilled meaning that a discrete Gauss Theorem is valid. This contradicts the perception in the hyperbolic community that stability issues for pure Galerkin scheme exist. In numerical simulations, we verify our theoretical analysis.

On the spatial formulation of discontinuous Galerkin methods for finite elastoplasticity

The telegraph equation is one of the important models in many physics and engineering. In this work, we discuss the high-order compact finite difference method for solving the two-dimensional second-order linear hyperbolic equation. By using a combined compact finite difference method for the spatial discretization, a high-order alternating direction implicit method (ADI) is proposed. The method is O(ź2 + h6) accurate, where ź, h are the temporal step size and spatial size, respectively. Von Neumann linear stability analysis shows that the method is unconditionally stable. Finally, numerical examples are used to illustrate the high accuracy of the new difference scheme.

Locally adapted meshes and polynomial degrees can greatly improve spectral element accuracy and applicability. A balancing domain decomposition by constraints (BDDC) preconditioner is constructed and analysed for both continuous (CG) and discontinuous (DG) Galerkin discretizations of scalar elliptic problems, built by nodal spectral elements with variable polynomial degrees. The DG case is reduced to the CG case via the auxiliary space method. The proposed BDDC preconditioner is proved to be scalable in the number of subdomains and quasi-optimal in both the ratio of local polynomial degrees and element sizes and the ratio of subdomain and element sizes. Several numerical experiments in the plane confirm the obtained theoretical convergence rate estimates, and illustrate the preconditioner performance for both CG and DG discretizations. Different configurations with locally adapted polynomial degrees are studied, as well as the preconditioner robustness with respect to discontinuities of the elliptic coefficients across subdomain boundaries. These results apply also to other dual-primal preconditioners defined by the same set of primal constraints, such as FETI-DP preconditioners.

A bulk metal oxide semiconductor field-effect transistor (MOSFET) is modelled with symmetric interior penalty discontinuous Galerkin (DG) method. The proposed framework is based on the drift-diffusion (DD) model and the Poisson equation. The Gummel algorithm is employed to decouple relevant equations. The potential and carrier distribution under a given bias are therefore obtained to demonstrate the feasibility of the proposed framework. Compared with the traditional continuous Galerkin (CG) method, this method is proven to achieve better numerical stability.

We revisit a method originally introduced by Werder et al. (in Comput. Methods Appl. Mech. Engrg., 190:6685–6708, 2001) for temporally discontinuous Galerkin FEMs applied to a parabolic partial differential equation. In that approach, block systems arise because of the coupling of the spatial systems through inner products of the temporal basis functions. If the spatial finite element space is of dimension D and polynomials of degree r are used in time, the block system has dimension (r + 1)D and is usually regarded as being too large when r > 1. Werder et al. found that the space-time coupling matrices are diagonalizable over for r ⩽100, and this means that the time-coupled computations within a time step can actually be decoupled. By using either continuous Galerkin or spectral element methods in space, we apply this DG-in-time methodology, for the first time, to second-order wave equations including elastodynamics with and without Kelvin–Voigt and Maxwell–Zener viscoelasticity. An example set of numerical results is given to demonstrate the favourable effect on error and computational work of the moderately high-order (up to degree 7) temporal and spatio-temporal approximations, and we also touch on an application of this method to an ambitious problem related to the diagnosis of coronary artery disease. Copyright © 2014 The Authors. International Journal for Numerical Methods in Engineering published by John Wiley & Sons Ltd.