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Register a new userBreaking spaces and forms for the DPG method and applications including Maxwell equations
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Discontinuous Petrov Galerkin (DPG) methods are made easily implementable using `broken test spaces, i.e., spaces of functions with no continuity constraints across mesh element interfaces. Broken spaces derivable from a standard exact sequence of first order (unbroken) Sobolev spaces are of particular interest. A characterization of interface spaces that connect the broken spaces to their unbroken counterparts is provided. Stability of certain formulations using the broken spaces can be derived from the stability of analogues that use unbroken spaces. This technique is used to provide a complete error analysis of DPG methods for Maxwell equations with perfect electric boundary conditions. The technique also permits considerable simplifications of previous analyses of DPG methods for other equations. Reliability and efficiency estimates for an error indicator also follow. Finally, the equivalence of stability for various formulations of the same Maxwell problem is proved, including the strong form, the ultraweak form, and a spectrum of forms in between.
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We introduce a cousin of the DPG method - the DPG* method - discuss their relationship and compare the two methods through numerical experiments.
This article introduces the DPG-star (from now on, denoted DPG$^*$) finite element method. It is a method that is in some sense dual to the discontinuous Petrov-Galerkin (DPG) method. The DPG methodology can be viewed as a means to solve an overdetermined discretization of a boundary value problem. In the same vein, the DPG$^*$ methodology is a means to solve an underdetermined discretization. These two viewpoints are developed by embedding the same operator equation into two different saddle-point problems. The analyses of the two problems have many common elements. Comparison to other methods in the literature round out the newly garnered perspective. Notably, DPG$^*$ and DPG methods can be seen as generalizations of $mathcal{L}mathcal{L}^ast$ and least-squares methods, respectively. A priori error analysis and a posteriori error control for the DPG$^*$ method are considered in detail. Reports of several numerical experiments are provided which demonstrate the essential features of the new method. A notable difference between the results from the DPG$^*$ and DPG analyses is that the convergence rates of the former are limited by the regularity of an extraneous Lagrange multiplier variable.
We develop and analyze a discontinuous Petrov--Galerkin method with optimal test functions (DPG method) for a shallow shell model of Koiter type. It is based on a uniformly stable ultraweak formulation and thus converges robustly quasi-uniformly. Numerical experiments for various cases, including the Scordelis--Lo cylindrical roof, elliptic and hyperbolic geometries, illustrate its performance. The built-in DPG error estimator gives rise to adaptive mesh refinements that are capable to resolve boundary and interior layers. The membrane locking is dealt with by raising the polynomial degree only of the tangential displacement trace variable.
We have developed a new fully anisotropic 3D FDTD Maxwell solver for arbitrary electrically and magnetically anisotropic media for piecewise constant electric and magnetic materials that are co-located over the primary computational cells. Two numerical methods were developed that are called non-averaged and averaged methods, respectively. The non-averaged method is first order accurate, while the averaged method is second order accurate for smoothly-varying materials and reduces to first order for discontinuous material distributions. For the standard FDTD field locations with the co-location of the electric and magnetic materials at the primary computational cells, the averaged method require development of the different inversion algorithms of the constitutive relations for the electric and magnetic fields. We provide a mathematically rigorous stability proof followed by extensive numerical testing that includes long-time integration, eigenvalue analysis, tests with extreme, randomly placed material parameters, and various boundary conditions. For accuracy evaluation we have constructed a test case with an explicit analytic solution. Using transformation optics, we have constructed complex, spatially inhomogeneous geometrical object with fully anisotropic materials and a large dynamic range of $underline{epsilon}$ and $underline{mu}$, such that a plane wave incident on the object is perfectly reconstructed downstream. In our implementation, the considerable increase in accuracy of the averaged method only increases the computational run time by 20%.
This paper introduces an ultra-weak space-time DPG method for the heat equation. We prove well-posedness of the variational formulation with broken test functions and verify quasi-optimality of a practical DPG scheme. Numerical experiments visualize beneficial properties of an adaptive and parabolically scaled mesh-refinement driven by the built-in error control of the DPG method.
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