Mimicking of Non-conservative Systems: Demonstration of a generalized method for applying the mimetic spectral element method to non-conservative systems

The mimetic spectral element method is a relatively young method in numerical solutions of partial differential equations and actions that describe physical systems. Its advantage is that it takes the geometrical structure of the problem into account which guarantees consistency of the numerical scheme and the conservation of relevant quantities...

Hybrid Dual Mimetic Spectral Element Method with a Novel Dual Grid

The mimetic spectral element method (MSEM) is a structure-preserving discretization scheme based on the Galerkin Method, which strongly constrains the topology relations by discretizing and reconstructing variables in specific function spaces in order to preserve certain critical structures of the PDE in the numerical solution. In studying the...

Development and Application of bundle-valued forms in hybrid mimetic spectral element method: Application to Linear elasticity

One of the novel methodologies in computational physics research is to use mimetic discretisation techniques. Among these, the mimetic spectral element method holds special promise as it not only has the benefits of mimetic methods but also the additional benefit of higher-order discretisations using higher polynomial degrees. These methods are...

Use of algebraic dual spaces in domain decomposition methods for Darcy flow in 3D domains

In this work we use algebraic dual spaces with a domain decomposition method to solve the Darcy equations. We define the broken Sobolev spaces and their finite dimensional counterparts. A global trace space is defined that connects the solution between the broken spaces. Use of algebraic dual spaces results in a sparse, metric-free...

The Lagrangian Mimetic Spectral Element Method: Solving (non-)Linear Advection Problems with a Mimetic Method

Advection is at the heart of fluid dynamics and is responsible for many interesting phenomena. Unfortunately, it is also the source of the non-linearity of fluid dynamics. As such, its numerical treatment is challenging and often suboptimal. One way to more effectively deal with advection is by using a Lagrangian formulation instead of the...

Hybridised Mimetic Discretisation and Variational Multiscale Theory for Advection-Dominated Problems: M.Sc. Thesis

Structure-preserving or mimetic discretisations are a class of advanced discretisation techniques derived by employing concepts from differential geometry. Such techniques can attain specific conservation properties at the discrete level such as conservation of mass, kinetic energy, etc when applied to conservation laws. However, like...

A Mimetic Spectral Element Implementation of the Maxwell Eigenvalue Problem

Mimetic formulations, also known as structure-preserving methods, are numerical schemes that preserve fundamental properties of the continuous differential operators at a discrete level. Additionally, they are well-known for satisfying constraints such as conservation of mass or momentum. <br/><br/>In the present work, a Mimetic Spectral Element...

Mimetic Spectral Element Method and Extensions toward Higher Computational Efficiency

Structure-conserving numerical methods that aim at preserving certain structures of the PDEs at the discrete level have been an interesting research topic for many decades. The mimetic spectral element method, a recently developed arbitrary order structure-preserving method on orthogonal or curvilinear meshes, has also been drawing increasingly...

A hybrid mimetic spectral element method for three-dimensional linear elasticity problems

We introduce a domain decomposition structure-preserving method based on a hybrid mimetic spectral element method for three-dimensional linear elasticity problems in curvilinear conforming structured meshes. The method is an equilibrium method which satisfies pointwise equilibrium of forces. The domain decomposition is established through...

Energy-conserving spectral element schemes based on Lagrangian dynamics: Investigating discrete energy conservation

Physical systems in the continuous domain are often solved using computer-aided software because of their complexity. Preserving the physical quantities from the continuous domain in the discrete domain is therefore of utmost importance. There is however a broad range of techniques that can accomplish the translation between the continuous and...

A high order hybrid mimetic discretization on curvilinear quadrilateral meshes for complex geometries

In this paper, we present a hybrid mimetic method which solves the mixed formulation of the Poisson problem on curvilinear quadrilateral meshes. The method is hybrid in the sense that the domain is decomposed into multiple disjoint elements and the interelement continuity is enforced using a Lagrange multiplier. The method is mimetic in the...

Preconditioning for Mimetic Spectral Element Method: A Preliminary Study Applied on Elliptic Equations

In order to solve linear system of equations obtained from numerical discretisation fast and accurate, preconditioning on the coefficient matrix is needed. In this thesis research, a preliminary study on preconditioning techniques suitable for Mimetic Spectral Element Method (MSEM) will be presented. The spectral limit change of some important...