Related Experiment Video
Updated: Oct 12, 2025
10:09
Operation of the Collaborative Composite Manufacturing CCM System
Published on: October 1, 2019
6.8K
-conforming finite element cochain complexes and commuting quasi-interpolation operators on Cartesian meshes.
Francesca Bonizzoni1, Guido Kanschat2
1Department of Mathematics, University of Augsburg, Universitätsstraße 14, 86159 Augsburg, Germany.
Calcolo
|November 22, 2021
Summary
A novel finite element cochain complex is introduced for Cartesian meshes, enabling stable, conforming finite element spaces. This method enhances accuracy in computational mathematics and numerical analysis by providing stable projectors.
Area of Science:
- Computational mathematics
- Numerical analysis
- Finite element methods
Background:
- Existing finite element methods may lack stability or conformity for certain differential operators.
- The need for robust numerical techniques for solving partial differential equations on Cartesian meshes is critical.
Purpose of the Study:
- To introduce a new finite element cochain complex suitable for Cartesian meshes in any dimension.
- To develop stable projectors that commute with the exterior derivative for conforming finite element spaces.
Main Methods:
- Construction of a finite element cochain complex based on the H(curl)-inner product.
- Utilizing a tensor product construction for creating H(curl)-stable projectors.
- Generalizing the finite element complex to arbitrary orders.
Main Results:
- The introduced complex yields H(curl)-conforming finite element spaces with exterior derivatives in L2.
- The tensor product construction successfully generates H(curl)-stable projectors.
- The finite element complex is shown to be generalizable to arbitrary orders.
Conclusions:
- The developed finite element cochain complex provides a stable and conforming framework for numerical computations on Cartesian meshes.
- This approach offers a generalized method for higher-order approximations, advancing numerical analysis techniques.
Keywords:
Commuting diagram propertyFinite element exterior calculusQuasi-interpolationTensor productde Rham complexMore Related Videos
Related Concept Videos
01:26Cartesian Form for Vector Formulation
823
The Cartesian form for vector formulation is a process to calculate the moment of force using the position and force vectors. The moment of force is defined as the cross-product of these vectors, making it a vector quantity. The Cartesian form of the position and force vectors involves unit vectors, which can be used to express the cross-product in determinant form.
823
01:25Coplanar Forces
4.7K
Consider an object upon which multiple forces are acting. If the lines of action of each force lie within the same plane, the system can be considered coplanar. The Cartesian vector form can be used to resolve each force into its respective components. For a coplanar system, the system will be in equilibrium if each component of the resultant force equals zero and the resultant force on the system is zero. If the sum of the forces is not equal to zero, then the object will not be in equilibrium...
4.7K
01:28Cartesian Vector Notation
1.0K
Cartesian vector notation is a valuable tool in mechanical engineering for representing vectors in three-dimensional space, performing vector operations such as determining the gradient, divergence, and curl, and expressing physical quantities such as the displacement, velocity, acceleration, and force. By using Cartesian vector notation, engineers can more easily analyze and solve problems in various areas of mechanical engineering, including dynamics, kinematics, and fluid mechanics. This...
1.0K
01:24Area Computation by the Alternative Coordinate Method
203
The alternative coordinate method, also known as the Shoelace Formula, is a technique for determining the area of a traverse using Cartesian coordinates. This method relies on the sequential arrangement of x and y coordinates for each point of the shape, ensuring accuracy and ease of application.In this approach, each corner's x and y coordinates are listed as fractions, with the x-coordinate as the numerator and the y-coordinate as the denominator. These coordinates are arranged sequentially...
203
01:20Mesh Analysis
1000
Mesh analysis is a valuable method for simplifying circuit analysis using mesh currents as key circuit variables. Unlike nodal analysis, which focuses on determining unknown voltages, mesh analysis applies Kirchhoff's voltage law (KVL) to find unknown currents within a circuit. This method is particularly convenient in reducing the number of simultaneous equations that need to be solved.
A fundamental concept in mesh analysis is the definition of meshes and mesh currents. A mesh is a closed...
A fundamental concept in mesh analysis is the definition of meshes and mesh currents. A mesh is a closed...
1000
01:21Couples: Scalar and Vector Formulation
366
One might wonder how the captain of a large ship can navigate through the ocean with just a turn of the steering wheel. The answer lies in the concept of two parallel forces that are equal in magnitude and opposite sense, creating a couple moment.
A couple moment is a rotational force that tends to rotate the steering wheel. The wheel's rotation can either be in a clockwise or anticlockwise direction. The right-hand rule is a helpful method for determining the direction of a couple moment....
A couple moment is a rotational force that tends to rotate the steering wheel. The wheel's rotation can either be in a clockwise or anticlockwise direction. The right-hand rule is a helpful method for determining the direction of a couple moment....
366